Among all the Hamiltonian systems, the alics integrable ones - those which have many conserved quantities - have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads to symplectic toric manifolds (part B of this book), which are examples of extremely symmetric Hamiltonian systems. Physics makes a surprising come-back in part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (part C of this book). Along the way, tools from many different areas of mathematics are brought to bear on the questions at hand, in particular, actions of Lie groups in symplectic and contact manifolds, the Delzant theorem, Morse theory, sheaves and \v{C}ech cohomology, and aspects of Calabi-Yau manifolds.

This book contains an expanded version of the lectures delivered by the authors at the CRM Barcelona in July 2001. It serves as an introduction to symplectic and contact geometry for graduate students and will be useful to research mathematicians interested in integrable systems.

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