This volume presents the papers based upon lectures given at the 1999 Seminaire de Mathematiques Superieurs held in Montreal. It includes contributions from many of the most active researchers in the field. This subject has been in a remarkably active state of development throughout the past three decades, resulting in new motivation for study in surprisingly different directions. Beyond the intrinsic interest in the study of integrable models of many-particle systems, spin chains, lattice and field theory models at both the classical and the quantum level, and completely solvable models in statistical mechanics, there have been new applications in relation to a number of other fields of current interest.These fields include theoretical physics and pure mathematics, for example the Seiberg-Witten approach to supersymmetric Yang-Mills theory, the spectral theory of random matrices, topological models of quantum gravity, conformal field theory, mirror symmetry, quantum cohomology, etc. This collection gives a nice cross-section of the current state of the work in the area of integrable systems which is presented by some of the leading active researchers in this field. The scope and quality of the articles in this volume make this a valuable resource for those interested in an up-to-date introduction and an overview of many of the main areas of study in the theory of integral systems.
On the chiral WZNW phase space, exchange r-matrices and Poisson-Lie groupoids by J. Balog, L. Feher, and L. Palla Loop groups, $R$-matrices and separation of variables by J. Harnad The geometry of generalised Hitchin systems by J. C. Hurtubise Determinant representation for form factors by V. E. Korepin Isomonodromic deformations in genus zero and one: Algebro-geometric solutions and Schlesinger transformations by D. A. Korotkin Quantum inverse scattering problem and correlation functions of integrable models by J.-M. Maillet Multiseparability and superintegrability for classical and quantum systems by W. Miller, Jr. Integrability and symmetry of the XXZ model by T. Miwa Characteristic systems on Poisson Lie groups and their quantization by N. Reshetikhin Special functions associated with Calogero-Moser type quantum systems by S. N. M. Ruijsenaars Backlund transformations and Baster's $Q$-operator by E. K. Sklyanin Universality of the distribution functions of random matrix theory by C. A. Tracy and H. Widom.