My talk will be an introductory lecture on the periodic orbits problem for Tonelli Hamiltonian systems. In a first part of the talk, I will recall the variational formulation of the problem, and its connections to Aubry-Mather theory and Weak KAM theory. I will then review some of the main known results on the multiplicity of periodic orbits with prescribed energy, and some of the celebrated open questions in the field, including the closed geodesics problem in Riemannian and Finsler manifolds.