This book is written in a pedagogical style intelligible for graduate students. It reviews recent progress in black-hole and wormhole theory and in mathematical cosmology within the framework of Einstein's field equations and beyond, including quantum effects. This collection of essays, written by leading scientists of long standing reputation, should become an indispensable source for future research.

The Euclidean approach to Quantum Gravity was initiated almost 15 years ago in an attempt to understand the difficulties raised by the spacetime singularities of classical general relativity which arise in the gravitational collapse of stars to form black holes and the entire universe in the Big Bang. An important motivation was to develop an approach capable of dealing with the nonlinear, non-perturbative aspects of quantum gravity due to topologically non-trivial spacetimes. There are important links with a Riemannian geometry. Since its inception the theory has been applied to a number of important physical problems including the thermodynamic properties of black holes, quantum cosmology and the problem of the cosmological constant. It is currently at the centre of a great deal of interest.This is a collection of survey lectures and reprints of some important lectures on the Euclidean approach to quantum gravity in which one expresses the Feynman path integral as a sum over Riemannian metrics. As well as papers on the basic formalism there are sections on Black Holes, Quantum Cosmology, Wormholes and Gravitational Instantons.

The Euclidean approach to Quantum Gravity was initiated almost 15 years ago in an attempt to understand the difficulties raised by the spacetime singularities of classical general relativity which arise in the gravitational collapse of stars to form black holes and the entire universe in the Big Bang. An important motivation was to develop an approach capable of dealing with the nonlinear, non-perturbative aspects of quantum gravity due to topologically non-trivial spacetimes. There are important links with a Riemannian geometry. Since its inception the theory has been applied to a number of important physical problems including the thermodynamic properties of black holes, quantum cosmology and the problem of the cosmological constant. It is currently at the centre of a great deal of interest.This is a collection of survey lectures and reprints of some important lectures on the Euclidean approach to quantum gravity in which one expresses the Feynman path integral as a sum over Riemannian metrics. As well as papers on the basic formalism there are sections on Black Holes, Quantum Cosmology, Wormholes and Gravitational Instantons.

The five lectures presented in this volume address very timely mathematical problems in relativity and cosmology. "Part I" is devoted to the initial value and evolution problems of the Einstein equations. Especially it deals with the Einstein-Yang-Mills-Boltzmann system, fluid models with finite or infinite conductivity, global evolution of a new (two-phase) model for gravitational collapse and the structure of maximal, asymptotically flat, vacuum solutions of the constraint equations which have the additional property of containing trapped surfaces. "Part II" focuses on geometrical-topological problems in relativity and cosmology: on the role of cosmic censorship for the global structure of the Einstein-Maxwell equations and on the mathematical structure of quantum conformal superspace.

This book is written in a pedagogical style intelligible for graduate students. It reviews recent progress in black-hole and wormhole theory and in mathematical cosmology within the framework of Einstein's field equations and beyond, including quantum effects. This collection of essays, written by leading scientists of long standing reputation, should become an indispensable source for future research.