Hamiltonian systems began as a mathematical approach to the study of mechanical systems. As the theory developed, it became clear that the systems that had a sufficient number of conserved quantities enjoyed certain remarkable properties. These are the completely integrable systems. In time, a rich interplay arose between integrable systems and other areas of mathematics, particularly topology, geometry, and group theory.This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations. Audin has included many examples and exercises. Most of the exercises build on the material in the text. None of the important proofs have been relegated to the exercises. Many of the examples are classical, rather than abstract. This book would be suitable for a graduate course in Hamiltonian systems.