This book presents some modern techniques in the theory of integrable systems, as variations on the theme of action-angle coordinates. The author starts with an able introduction to integrable systems, including symplectic manifolds, Hamiltonian systems, completely integrable systems, and geodesic flows. She continues with action-angle variables, integrability and Galois groups (including the Arnold-Liouville theorem, as revisited through differentia Galois theory), Lax closing with an introduction to Lax equations as they apply to the Arnold-Liouville theorem. The author includes exercises and two appendices, the first on what one needs to know about differential Galois theory and algebraic curves, making this a suitable candidate for self-study as well as a classroom text.