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.