  We prove the action-angle theorem in the general, and most natural, context
of integrable systems on Poisson manifolds, thereby generalizing the classical
proof, which is given in the context of symplectic manifolds. The topological
part of the proof parallels the proof of the symplectic case, but the rest of
the proof is quite different, since we are naturally led to using the calculus
of polyvector fields, rather than differential forms; in particular, we use in
the end a Poisson version of the classical Caratheodory-Jacobi-Lie theorem,
which we also prove. At the end of the article, we generalize the action-angle
theorem to the setting of non-commutative integrable systems on Poisson
manifolds.
