Differential equations, including Hamiltonian equations, are customarily classified into integrable and nonintegrable. “When, however, one attempts to formulate a precise definition of integrability, many possibilities appear, each with a certain intrinsic theoretic interest” (D. Birkhoff, Dynamical Systems). In this section we give a brief survey of various approaches to integrability of Hamiltonian systems, “not forgetting the dictum of Poincare, that a system of differential equations is only more or less integrable” (ibid.).