This book is another volume in the successful subseries "e;Dynamical Systems"e; of the Encyclopaedia of Mathematical Sciences. It focuses on new developments in the field of integrable systems and it provides a unique and comprehensive survey on new differential geometric and Lie-algebraic methods. Since other literature on these topics is often rather unreadable, this volume will be an indispensable guide to the current research which no mathematician or theoretical physicist who is interested in this field can do without.