  In this paper we develop a general conceptual approach to the problem of
existence of action-angle variables for dynamical systems, which establishes
and uses the fundamental conservation property of associated torus actions:
anything which is preserved by the system is also preserved by the associated
torus actions. This approach allows us to obtain, among other things: a) the
shortest and most conceptual easy to understand proof of the classical
Arnold--Liouville--Mineur theorem; b) basically all known results in the
literature about the existence of action-angle variables in various contexts
can be recovered in a unifying way, with simple proofs, using our approach; c)
new results on action-angle variables in many different contexts, including
systems on contact manifolds, systems on presymplectic and Dirac manifolds,
action-angle variables near singularities, stochastic systems, and so on. Even
when there are no natural action variables, our approach still leads to useful
normal forms for dynamical systems, which are not necessarily integrable.
