Locally Lagrangian symplectic and poisson manifolds

We discuss symplectic manifolds where, locally, the structure is that encountered in Lagrangian dynamics. Examples and characteristic properties are given. Then, we refer to the computation of the Maslov classes of a Lagrangian submanifold. Finally, we indicate the generalization of this type of symplectic structures to Poisson manifolds. The paper is the text of a lecture presented at the Conference "Poisson 2000" held at CIRM, Luminy, France, between June 26 and June 30, 2000. It reviews results contained in the author's papers [9, 12, 13] as well as in papers by other authors.