The majority of books dealing with prospects for interstellar flight tackle the problem of the propulsion systems that will be needed to send a craft on an interstellar trajectory. The proposed book looks at two other, equally important aspects of such space missions, and each forms half of this two part book.

Part 1 looks at the ways in which it is possible to exploit the focusing effect of the Sun as a gravitational lens for scientific missions to distances of 550 AU and beyond into interstellar space. The author explains the mechanism of the Sun as a gravitational lens, the scientific investigations which may be carried out along the way to a distance of 550 AU (and at the 550 AU sphere itself), the requirements for exiting the Solar System at the highest speed and a range of project ideas for missions entering interstellar space.

Part 2 of the book deals with the problems of communicating between an interstellar spaceship and the Earth, especially at very high speeds. Here the author assesses a range of mathematical tools relating to the Karhunen-Loeve Transform (KLT) for optimal telecommunications, technical topics that may one day enable humans flying around the Galaxy to keep in contact with the Earth. This part of the book opens with a summary of the author's 2003 Pe ek Lecture presented at the IAC in Bremen, which introduces the concept of KLT for engineers and 'newcomers' to the subject. It is planned to include a DVD containing the full mathematical derivations of the KLT for those interested in this important mathematical tool whilst the text itself will contain the various results without outlines of the mathematical proofs. Astronautical engineers will thus be able to see the application of the results without getting bogged down in the mathematics."

Relativistic quantum electrodynamics, which describes the electromagneticinteractions of electrons and atomic nuclei, provides the basis for modeling the electronic structure of atoms, molecules and solids and of their interactions with photons and other projectiles. The theory underlying the widely used GRASP relativistic atomic structure program, the DARC electron-atom scattering code and the new BERTHA relativistic molecular structure program is presented in depth, together with computational aspects relevant to practical calculations. Along with an understanding of the physics and mathematics, the reader will gain some idea of how to use these programs to predict energy levels, ionization energies, electron affinities, transition probabilities, hyperfine effects and other properties of atoms and molecules.

This mathematically-oriented introduction takes the point of view that students should become familiar, at an early stage, with the physics of relativistic continua and thermodynamics within the framework of special relativity. Therefore, in addition to standard textbook topics such as relativistic kinematics and vacuum electrodynamics, the reader will be thoroughly introduced to relativistic continuum and fluid mechanics. There is emphasis on the 3+1 splitting technique.

This book is a comprehensive reference on differential geometry. It shows that Maxwell, Dirac and Einstein fields, which were originally considered objects of a very different mathematical nature, have representatives as objects of the same mathematical nature. The book also analyzes some foundational issues of relativistic field theories. All calculation procedures are illustrated by many exercises that are solved in detail.

Here is a systematic approach to such fundamental questions as: What mathematical structures does Einstein-Weyl causality impose on a point-set that has no other previous structure defined on it? The author proposes an axiomatization of the physics inspired notion of Einstein-Weyl causality and investigating the consequences in terms of possible topological spaces. One significant result is that the notion of causality can effectively be extended to discontinuum.

The formation of galaxies is one of the greatest puzzles in astronomy, the solution is shrouded in the depths of space and time, but has profound implications for the universe we observe today. This book discusses the beginnings of the process from cosmological observations and calculations. It examines the different theories of galaxy formation and shows where each theory either succeeds or fails in explaining what we actually observe. In addition, the book looks ahead to what we may expect to uncover about the epoch of galaxy formation from the new and upcoming generations of telescopes and technology.

All physicists would agree that one of the most fundamental problems of the 21st century physics is the dimensionality of the world. In the four-dimensional world of Minkowski (or Minkowski spacetime) the most challenging problem is the nature of the temporal dimension. In Minkowski spacetime it is merely one of the four dimensions, which means that it is entirely given like the other three spacial dimensions. If the temporal dimension were not given in its entirety and only one constantly changing moment of it existed, Minkowski spacetime would be reduced to the ordinary three-dimensional space.

But if the physical world, represented by Minkowski spacetime, is indeed four-dimensional with time being the fourth dimension, then such a world is drastically different from its image based on our perceptions. Minkowski four-dimensional world is a block Universe, a frozen world in which nothing happens since all moments of time are given at once', which means that physical bodies are four-dimensional worldtubes containing the whole histories in time of the three-dimensional bodies of our everyday experience. The implications of a real Minkowski world for physics itself and especially for our world view are enormous.

The main focus of this volume is the question: is spacetime nothing more than a mathematical space (which describes the evolution in time of the ordinary three-dimensional world) or is it a mathematical model of a real four-dimensional world with time entirely given as the fourth dimension? It contains fourteen invited papers which either directly address the main question of the nature of spacetime or explore issues related to it."

Thisbookisaneditedversionofthelecturesdeliveredduringthe1stAegean SummerSchoolonCosmology,heldonSamosisland,Greece,inSeptember 21-29,2001,andorganizedjointlybytheDepartmentofMathematics,U- versity of the Aegean and the Department of Physics, National Technical UniversityofAthens. Cosmology,thescienceoftheuniverse,standsatthecrossroadsofmany ?eldsofphysicsandmathematicsandpresentsuswithchallengingproblems of many forms. Although there are by now many textbooks discussing the subjectatmanylevels,itistruethatnosinglebookhasthecharacteristics wehadinmindwheneditingthisvolume. Wehavetriednottoproducea proceedingsvolumebutmoreamultiauthoredtextbookwhichcouldserveas areferencesourceofcurrentideasincosmology. Webelievethisbookcovers atanintroductorylevelmostoftheissueswhichareconsideredimportant inmoderncosmologicalresearchandcanbereadbyagraduatestudentor researcherwhowishestoacquireareasonableknowledgeofcosmologythat will,wehope,continuetobeofvalueforyearstocome. The 1st Aegean School on Cosmology, and consequently this book, - camepossiblewiththekindsupportofmanypeopleandorganizations. We received ?nancial support from the following sources and this is gratefully acknowledged: the Municipality of Karlovassi, the North Aegean Regional Secretariat, the Prefecture of Samos, the Ministry of the Aegean, and the NationalBankofGreece. TheadministrativesupportoftheSchoolwastakenupwithgreatcare byMrs. EvelynPappaandMantoKatsianiandwewouldliketothankthem bothfortheirkinde?ortstoresolvemanyissueswhicharosebefore,during andaftertheSchool. WeacknowledgethehelpofMr. NectariosBenekoswho designedandmaintainedthewebsiteoftheSchool. Last,butnotleast,wearegratefultothesta?ofSpringer-Verlag,resp- siblefortheLectureNotesinPhysics,whoseabilitiesandhelpcontributed greatlytothe?neappearanceofthisbook. Karlovassi,Samos, SpirosCotsakis March2002 EleftheriosPapantonopoulos TableofContents PartI HistoryandOverview 1 IsNatureGeneric? SpirosCotsakis,PeterG. L. Leach...3 1. 1 Introduction...3 1. 2 PrinciplesofCosmologicalModelling...4 1. 2. 1 Spacetimes...4 1. 2. 2 TheoriesofGravity...5 1. 2. 3 MatterFields...6 1. 3 Cosmologies...6 1. 4 CosmologicalProblems...8 1. 4. 1 TheSingularityProblem...8 1. 4. 2 TheProblemofCosmicTopology...9 1. 4. 3 TheProblemofAsymptoticStates...9 1. 4. 4 GravityTheoriesandtheEarlyUniverse...11 1. 5Outlook ...12 References...14 2 EvolutionofIdeasinModernCosmology AndreasParaskevopoulos...16 2. 1 Introduction...16 2. 2 TheBeginningsofModernCosmology(1917-1950)...17 2. 3 Cosmology1950-1970:HotBigBang, SingularitiesandQuantumApproach...20 2. 4 Cosmology1970-Thisbookisaneditedversionofthelecturesdeliveredduringthe1stAegean SummerSchoolonCosmology,heldonSamosisland,Greece,inSeptember 21-29,2001,andorganizedjointlybytheDepartmentofMathematics,U- versity of the Aegean and the Department of Physics, National Technical UniversityofAthens. Cosmology,thescienceoftheuniverse,standsatthecrossroadsofmany ?eldsofphysicsandmathematicsandpresentsuswithchallengingproblems of many forms. Although there are by now many textbooks discussing the subjectatmanylevels,itistruethatnosinglebookhasthecharacteristics wehadinmindwheneditingthisvolume. Wehavetriednottoproducea proceedingsvolumebutmoreamultiauthoredtextbookwhichcouldserveas areferencesourceofcurrentideasincosmology. Webelievethisbookcovers atanintroductorylevelmostoftheissueswhichareconsideredimportant inmoderncosmologicalresearchandcanbereadbyagraduatestudentor researcherwhowishestoacquireareasonableknowledgeofcosmologythat will,wehope,continuetobeofvalueforyearstocome. The 1st Aegean School on Cosmology, and consequently this book, - camepossiblewiththekindsupportofmanypeopleandorganizations. We received ?nancial support from the following sources and this is gratefully acknowledged: the Municipality of Karlovassi, the North Aegean Regional Secretariat, the Prefecture of Samos, the Ministry of the Aegean, and the NationalBankofGreece. TheadministrativesupportoftheSchoolwastakenupwithgreatcare byMrs. EvelynPappaandMantoKatsianiandwewouldliketothankthem bothfortheirkinde?ortstoresolvemanyissueswhicharosebefore,during andaftertheSchool. WeacknowledgethehelpofMr. NectariosBenekoswho designedandmaintainedthewebsiteoftheSchool. Last,butnotleast,wearegratefultothesta?ofSpringer-Verlag,resp- siblefortheLectureNotesinPhysics,whoseabilitiesandhelpcontributed greatlytothe?neappearanceofthisbook. Karlovassi,Samos, SpirosCotsakis March2002 EleftheriosPapantonopoulos TableofContents PartI HistoryandOverview 1 IsNatureGeneric? SpirosCotsakis,PeterG. L. Leach...3 1. 1 Introduction...3 1. 2 PrinciplesofCosmologicalModelling...4 1. 2. 1 Spacetimes...4 1. 2. 2 TheoriesofGravity...5 1. 2. 3 MatterFields...6 1. 3 Cosmologies...6 1. 4 CosmologicalProblems...8 1. 4. 1 TheSingularityProblem...8 1. 4. 2 TheProblemofCosmicTopology...9 1. 4. 3 TheProblemofAsymptoticStates...9 1. 4. 4 GravityTheoriesandtheEarlyUniverse...11 1. 5Outlook ...12 References...14 2 EvolutionofIdeasinModernCosmology AndreasParaskevopoulos...16 2. 1 Introduction...16 2. 2 TheBeginningsofModernCosmology(1917-1950)...17 2. 3 Cosmology1950-1970:HotBigBang, SingularitiesandQuantumApproach...20 2. 4 Cosmology1970-1990:Chaotic,In?ationary, QuantumandAlternative...22 2. 5ConclusionsandOutlook ...25 References...26 VIII TableofContents PartII MathematicalCosmology 3ConstraintsandEvolutioninCosmology YvonneChoquet-Bruhat,JamesW. York...29 3. 1 Introduction...29 3. 2 MovingFrameFormulas...30 3. 2. 1 FrameandCoframe...30 3. 2. 2 Metric...31 3. 2. 3 Connection...31 3. 2. 4 Curvature ...32 3. 3 (n+1)-SplittingAdaptedtoSpaceSlices ...33 3. 3. 1 De?nitions...33 3. 3. 2 StructureCoe?cients...34 3. 3. 3 SplittingoftheConnection ...

Gravitational waves (GWs) are a hot topic and promise to play a central role in astrophysics, cosmology, and theoretical physics. Technological developments have led us to the brink of their direct observation, which could become a reality in the coming years. The direct observation of GWs will open an entirely new field: GW astronomy. This is expected to bring a revolution in our knowledge of the universe by allowing the observation of previously unseen phenomena, such as the coalescence of compact objects (neutron stars and black holes), the fall of stars into supermassive black holes, stellar core collapses, big-bang relics, and the new and unexpected. With a wide range of contributions by leading scientists in the field, Gravitational Waves covers topics such as the basics of GWs, various advanced topics, GW detectors, astrophysics of GW sources, numerical applications, and several recent theoretical developments. The material is written at a level suitable for postgraduate students entering the field.

REFLECTIONS ON SPACETIME - FOUNDATIONS, PHILOSOPHY AND HISTORY During the academic year 1992/93, an interdisciplinary research group constituted itself at the Zentrum fUr interdisziplinare Forschung (ZiF) in Bielefeld, Germany, under the title 'Semantical Aspects of Spacetime Theories', in which philosophers and physicists worked on topics in the interpretation and history of relativity theory. The present issue consists of contributions resulting from material presented and discussed in the group during the course of that year. The scope of the papers ranges from rather specialised issues arising from general relativity such as the problem of referential indeterminacy, to foundational questions regarding spacetime in the work of Carnap, Weyl and Hilbert. It is well known that the General Theory of Relativity (GTR) admits spacetime models which are 'exotic' in the sense that observers could travel into their own past. This poses a number of problems for the physical interpretation of GTR which are also relevant in the philosophy of spacetime. It is not enough to exclude these exotic models simply by stating that we live in a non-exotic universe, because it might be possible to "operate time machines" by actively changing the topology of the future part of spacetime. In his contribution, Earman first reviews the attempts of physicists to prove "chronology protection theorems" (CPTs) which exclude the operation of time machines under reasonable assumptions.

The Evolution of Complexity is addressed to a broad audience of academics and researchers from different disciplines, who are interested in the picture of our world emerging from the new sciences of complexity. This book reviews the new concepts proposed by the diverse theories of evolution, self-organisation, general systems, cybernetics, and the complex adaptive systems' approach pioneered by the Santa Fe institute. The thread which holds everything together is the growth of complexity during the history of the universe: from elementary particles, via atoms, molecules, living cells, multicellular organisms, plants, and animals to human beings, and societies. The different sections of the book discuss the foundations and philosophy of complexity evolution, its mathematical and computer models, its explanation of self-organising and living systems, the insights it provides into the origin of mind, language and culture, and its practical applications in areas such as management and system design.

Causal relations, and with them the underlying null cone or conformal structure, form a basic ingredient in all general analytical studies of asymptotically flat space-time. The present book reviews these aspects from the analytical, geometrical and numerical points of view. Care has been taken to present the material in a way that will also be accessible to postgraduate students and nonspecialist reseachers from related fields.

This book contains the expanded lecture notes of the 32nd Saas-Fee Advanced Course. The three contributions present the central themes in modern research on the cold universe, ranging from cold objects at large distances to the physics of dust in cold clouds.

Is time, even locally, like the real line? Multiple structures of time, implicit in physics, create a consistency problem. A tilt in the arrow of time is suggested as the most conservative hypothesis which provides approximate consistency within physics and with topology of mundane time. Mathematically, the assumed constancy of the velocity of light (needed to measure time) implies functional differential equations of motion, that have both retarded and advanced deviating arguments with the hypothesis of a tilt. The novel features of such equations lead to a nontrivial structure of time and quantum-mechanical behaviour. The entire argument is embedded in a pedagogical exposition which amplifies, corrects, and questions the conventionally accepted approach. The exposition includes historical details and explains, for instance, why the entropy law is inadequate for time asymmetry, and why notions such as time asymmetry (hence causality) may be conceptually inadequate. The first three parts of the book are especially suited as supplementary reading material for undergraduate and graduate students and teachers of physics. The new ideas are addressed to researchers in physics and philosophy of science concerned with relativity and the interpretation of quantum mechanics.

Dark matter research is one of the most fascinating and active fields among current high-profile scientific endeavours. It holds the key to all major breakthroughs to come in the fields of cosmology and astroparticle physics. The present volume is particularly concerned with the sources and the detection of dark matter and dark energy in the universe and will prove to be an invaluable research tool for all scientists who work in this field.

Bell's Theorem and its associated implications for the nature of the physical world remain topics of great interest. For this reason many meetings have been recently held on the interpretation of quantum theory and the implications of Bell's Theorem. Generally these meetings have been held primarily for quantum physicists and philosophers of science who have been or are actively working on the topic. Nevertheless, other philosophers of science, mathematicians, engineers as well as members of the general public have increasingly taken interest in Bell's Theorem and its implications. The Fall Workshop held at George Mason University on October 21 and 22, 1988 and titled "Bell's Theorem, Quantum Theory and Conceptions of the Universe" was of a more general scope. Not only it attracted experts in the field, it also covered other topics such as the implications of quantum non-locality for the nature of consciousness, cosmology, the anthropic principle, etc. topics usually not covered in previous meetings of this kind. The meeting was attended by more than one hundred ten specialists and other interested people from all over the world. The purpose of the meeting was not to provide a definitive answer to the general questions raised by Bell's Theorem. It is likely that the debate will go on for quite a long time. Rather, it was meant to contribute to the important dialogue between different disciplines.

Back in1954,a paper[2] by Bondi and Gold was to pick upona much olderqu- tion and raise anew one that would trigger another longdebate. The old question hadbeenaroundsince the beginning of the twentiethcentury, whenBorn?rstraised it[1] and others followed suit. This was the question of whethera uniformly acc- erated charge (in?at spacetime) would radiateelectromagnetic energy. The new question arose from the claim by Bondi and Gold that (inthe contextof general relativity now)a static charge ina static gravitational ?eld cannot radiateenergy. If this were the case,thenaparticular version of the equivalence principle would thereby be contradicted. This book reviews the problem discovered by Bondi and Gold and discusses the ensuingdebate ascarried on by Fulton and Rohrlich [3], DeWitt and Brehme [4], Mould [5], Boulware [6], andParrott [7].Various solutionshave been proposed by the above (and otherswhoare not discussed here). One of the aims here will be to putforward arather different solution to Bondi and Gold's radiation problem. So eventhough the paperscited are discussed to a large extent in chronological order, the reason for writing this is not justto produce an historical reference. Andeven though the version of general relativity applied hereis entirely consensual, every one of these papersis criticised on at leastoneimportant count, soI suspectthat the resultas a whole should not be described asconsensual.

This is the third volume in a series of books on the general topics of Supers- metric Mechanics, with the ?rst and second volumes being published as Lecture Notes in Physics Vol. 698, Supersymmetric Mechanics - Vol. 1: Supersymmetry, Noncommutativity and Matrix Models (ISBN: 3-540-33313-4), and Lecture Notes in Physics Vol. 701, Supersymmetric Mechanics - Vol. 2: The Attractor Mechanism and Space Time Singularities (ISBN: 3-540-34156-0). The aim of this ongoing collection is to provide a reference corpus of suitable, introductory material to the ?eld, by gathering the signi?cantly expanded and edited versions of all tutorial lectures, given over the years at the well-established annual INFN-Laboratori Nazionali di Frascati Winter School on the Attractor Mechanism, directed by myself. The present set of notes results again from the participation and dedication of prestigious lecturers, such as Iosif Bena, Sergio Ferrara, Renata Kallosh, Per Kraus, Finn Larsen, and Boris Pioline. As usual, the lectures were subsequently carefully edited and reworked, taking into account the extensive follow-up discussions. The present volume emphasizes topics of great recent interest, namely general concepts of attractors in supersymmetric gravity and black holes.

Quasars, and the menagerie of other galaxies with "unusual nuclei," now collectively known as Active Galactic Nuclei or AGN, have, in one form or another, sparked the interest of astronomers for over 60 years. The only known mechanism that can explain the staggering amounts of energy emitted by the innermost regions of these systems is gravitational energy release by matter falling towards a supermassive black hole --- a black hole whose mass is millions to billions of times the mass of our Sun. AGN emit radiation at all wavelengths. X-rays originating at a distance of a few times the event horizon of the black hole are the emissions closest to the black hole that we can detect; thus, X-rays directly reveal the presence of active supermassive black holes. Oftentimes, however, the supermassive black holes that lie at the centers of AGN are cocooned in gas and dust that absorb the emitted low energy X-rays and the optical and ultraviolet light, hiding the black hole from view at these wavelengths. Until recently, this low-energy absorption presented a major obstacle in observational efforts to map the accretion history of the universe. In 1999 and 2000, the launches of the Chandra and XMM-Newton X-ray Observatories finally broke the impasse. The impact of these observatories on X-ray astronomy is similar to the impact that the Hubble Space Telescope had on optical astronomy. The astounding new data from these observatories have enabled astronomers to make enormous advances in their understanding of when accretion occurs."

Each contribution is an article in itself, and great effort has been made by the authors to be lucid and not too technical. A few brief highlights of the round-table discussions are given between the chapters.

Topics include: Quantum non-locality, the measurement problem, quantum insights into relativity, cosmology and thermodynamics, and possible bearings of quantum mechanics to biology and consciousness. Authors include Yakir Aharanov and Anton Zeilinger, plus Nobel laureates Anthony J. Leggett (2003) and Gerardus t Hooft (1999).

Foreword written by Sir Roger Penrose, best-selling author (The Emperor's New Mind) and world-renowned mathematical physicist.

The Workshop "Science with the VLT in the ELT Era" held in Garching from 8th to 12th October 2007 was organised by ESO, with support from its Scienti c and Technical Committee, to provide a forum for the astronomical community to debate the long term future of ESO's Very Large Telescope (VLT) and its interferometric mode (VLTI). In particular it was considered useful for future planning to evaluate how its science use may evolve over the next decade due to competition and/or synergy with new facilities such as ALMA, JWST and, hopefully, at least one next generation 30-40 m extremely large telescope whose acronym appears in the title to symbolise this wider context. These discussions were also held in the fresh light of the Science Vision recently developed within ASTRONET as the rst step towards a 20 year plan for implementing astronomical facilities-the rst such attempt within Europe. Speci c ideas and proposals for new, second generation VLT/I instruments were also solicited following a tradition set by several earlier Workshops held since the start of the VLT development. The programme consisted of invited talks and reviews and contributed talks and posters. Almost all those given are included here although, unfortunately not the several lively but constructive discussion sessions.

This book looks at answers to the biggest questions in astronomy - the questions of how the planets, stars, galaxies and the universe were formed. Over the last decade, a revolution in observational astronomy has produced possible answers to three of these questions. This book describes this revolution. The one question for which we still do not have an answer is the question of the origin of the universe. In the final chapter, the author looks at the connection between science and philosophy and shows how new scientific results have laid the groundwork for the first serious scientific studies of the origin of the universe.

The contemporary theoretical physics consists, by and large, of two independent parts. The rst is the quantum theory describing the micro-world of elementary p- ticles, the second is the theory of gravity that concerns properties of macroscopic systems such as stars, galaxies, and the universe. The relativistic theory of gr- itation which is known as general relativity was created, at the beginning of the last century, by more or less a single man from pure idea combinations and bold guessing. The task was to "marry" the theory of gravity with the theory of special relativity. The rst attempts were aimed at considering the gravitational potential as a eld in Minkowski space-time. All those attempts failed; it took 10 years until Einstein nally solved the problem. The dif culty was that the old theory of gravity as well as the young theory of special relativity had to be modi ed. The next 50 years were dif cult for this theory because its experimental basis remained weak and its complicated mathematical structure was not well understood. However, in the subsequent period this theory ourished. Thanks to improvements in the te- nology and to the big progress in the methods of astronomical observations, the amount of observable facts to which general relativity is applicable was consid- ably enlarged. This is why general relativity is, today, one of the best experimentally tested theories while many competing theories could be disproved. Also the conc- tual and mathematical fundamentals are better understood now.

In 1997, contrary to the ruling paradigm which was that of a dark matter ?lled, decelerating universe, my work pointed to a dark energydriven- celerating universe with a small cosmological constant. Moreover, the many supposedly accidental Large Number relations in cosmology, including the mysterious Weinbergformula were now deduced from the theory. Obser- tionalcon?rmationforthisscenariocamein1998, whiledarkenergyitselfwas ?nally recon?rmed in 2003, thanks to the Wilkinson Microwave Anisotropy Probe and the Sloan Digital Sky Survey. The 1997, and subsequent work was the consequence of mainly three cons- erations: dark energy or the well known Zero Point Field, fuzzy spacetime and ?uctuations. Indeed String Theory and Quantum Gravity approaches have had to discard the smooth spacetime of General Relativity and Qu- tum Field Theory, in a quest for a uni?ed description of these two pillars of twentieth century physics. This book is the result of some seventy ?ve papers published in international journals, andpartlyanearlierbook,"TheChaoticUniverse: FromthePlanck to the Hubble Scale" (Nova Science, New York, 2001), as also several lectures delivered in Universities and institutes in the United States, Canada and - rope. It describes how, in a simple and somewhat conventional framework, an underpinning of Planck scale oscillators in the ubiquitous Zero Point Field or dark energy leads to a uni?ed description of phenomena involving elementary particles and the cosmos. In particular, apart from the cosmology mentioned above, these considerations lead to a uni?ed description of all interactions, includinggravitation, thoughinanextended gauge ?eld treatment.

Soliton theory is an important branch of applied mathematics and mathematical physics. An active and productive field of research, it has important applications in fluid mechanics, nonlinear optics, classical and quantum fields theories etc. This book presents a broad view of soliton theory. It gives an expository survey of the most basic ideas and methods, such as physical background, inverse scattering, Backl nd transformations, finite-dimensional completely integrable systems, symmetry, Kac-moody algebra, solitons and differential geometry, numerical analysis for nonlinear waves, and gravitational solitons. Besides the essential points of the theory, several applications are sketched and some recent developments, partly by the authors and their collaborators, are presented.