This book is the third in a series of lectures of the S' eminaire Poincar' e,whichis directed towards a large audience of physicists and of mathematicians. The goal of this seminar is to provide up to date information about general topics of great interest in physics. Both the theoretical and experimental aspects are covered, with some historical background. Inspired by the Bourbaki seminar in mathematics in its organization, hence nicknamed "Bourbaphi", this Poincar' e Seminar is held twice a year at the Institut Henri Poincar' ein Paris,with contri- tions prepared in advance. A particular care is devoted to the pedagogical nature of the presentation so as to ful?ll the goal of being readable by a large audience of scientists. ThisvolumecontainsthesixthsuchSeminar,heldin2004.Itisdevotedtothe Quantum Hall E?ect. After a historical and general presentation by Nobel prize Klaus von Klitzing, discoverer of this e?ect, the volume proceeds with reviews on the mathematics and physics of both the integer and fractional case, and includes up to date presentations of the tunneling and metrology experiments related to the Quantum Hall E?ect. We hopethat the publicationof this serieswill servethe community of phy- cists and mathematicians at professional or graduate student level. ' We thank the Commissariat 'al'Energie Atomique (Division des Sciences de laMati' ere),theCentreNationaldelaRechercheScienti?que(SciencesPhysiqueet Math' ematiques), and the Daniel Iagolnitzer Foundation for sponsoring the Se- nar. Special thanks are due to Chantal Delongeas for the preparation of the m- uscript.

Recent years have shown important and spectacular convergences between techniques traditionally used in theoretical physics and methods emerging from modern mathematics (combinatorics, probability theory, topology, algebraic geometry, etc). These techniques, and in particular those of low-dimensional statistical models, are instrumental in improving our understanding of emerging fields, such as quantum computing and cryptography, complex systems, and quantum fluids. This book sets these issues into a larger and more coherent theoretical context than is currently available. For instance, understanding the key concepts of quantum entanglement (a measure of information density) necessitates a thorough knowledge of quantum and topological field theory, and integrable models. To achieve this goal, the lectures were given by international leaders in the fields of exactly solvable models in low dimensional condensed matter and statistical physics.