We first introduce integrable Hamiltonian systems on symplectic manifolds. We show that if a Hamiltonian system on a two–dimensional phase space has all of its orbits closed then we can modify the Hamiltonian by a diffeomorphism to ensure all the orbits have the same period. The rest of the chapter explains how to generalise this to Hamiltonian systems with more degrees of freedom, culminating in the Arnold–Liouville theorem, which underpins everything else in the book.