This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. While infinite-dimensional groups often exhibit very peculiar features, this book describes unifying geometric ideas of the theory and gives numerous illustrations and examples, ranging from the classification of the Virasoro coadjoint orbits to knot theory, from optimal mass transport to moduli spaces of flat connections on surfaces. The text includes many exercises and open questions, and it is accessible to both students and researchers in Lie theory, geometry, and Hamiltonian systems.

B.Khesin's areas of research are infinite-dimensional Lie groups, integrable systems, Poisson geometry, and topological hydrodynamics. Together with Vladimir Arnold he is the author of the monograph on "Topological methods in hydrodynamics", which has become a standard reference in mathematical fluid dynamics. He was a Sloan research fellow in 1997-1999 and a Clay Mathematics Institute book fellow in 2006-2007, as well as an Andre-Aizenstadt prize recepient in 1998. R.Wendt's fields of research include the geometry and representation theory of infinite dimensional Lie groups and algebras, related geometric structures, and mathematical physics. He is also interested in mathematical finance and 'real world' applications of financial modelling

Preface.- Introduction.- I Preliminaries.- II Infinite-dimensional Lie Groups: Their Geometry, Orbits and Dynamical Systems.- III Applications of Groups: Topological and Holomorphic Gauge Theories.- Appendices.- A1 Root Systems.- A2 Compact Lie Groups.- A3 Krichever-Novikov Algebras.- A4 Kähler Structures on the Virasoro and Loop Group Coadjoint Orbits.- A5 Metrics and Diameters of the Group of Hamiltonian Diffeomorphisms.- A6 Semi-Direct Extensions of the Diffeomorphism Group and Gas Dynamics.- A7 The Drinfeld-Sokolov Reduction.- A8 Surjectivity of the Exponential Map on Pseudo-Differential Symbols.- A9 Torus Actions on the Moduli Space of Flat Connections.- Bibliography.- Index