Recent advances in the study of the existence of periodic orbits of Hamiltonian systems -- The direct method in the study of periodic solutions of Hamiltonian systems with prescribed period -- Periodic solutions of Hamiltonian systems having prescribed minimal period -- Duality in non convex variational problems -- La théorie de la seconde variation et le problème linéaire quadratique -- Méthodes asymptotiques dans l'étude de systèmes hamiltoniens non autonomes -- Numerical research of periodic solutions for a Hamiltonian system.