In my talk I will give two motivations for the development of Aubry-Mather theory (AMT), one coming from Hamiltonian dynamics and one coming from the calculus of variations. AMT lies at the junction of both fields and gives insight into the phenomenon encountered in Hamiltonian dynamics and the calculus of variations. In the second part I will explain how to generalize the theory to Lorentzian geometry. In opposition to the positive definite case, some assumptions are needed. This defines a new class of compact spacetimes with interesting properties. If time permits I will give the statements of some results obtained in the Lorentzian AMT.