Periodic solutions of Hamiltonian PDEs

This paper surveys some recent results about periodic solutions for Hamiltonian PDEs, which involve the small denominators difficulty. We considers in particular periodic orbits for completely resonant nonlinear wave equations and "Birkhoff-Lewis'' orbits with applications to a semilinear beam equation and to the nonlinear Schrödinger equation. The common key technique is a variational Lyapunov-Schmidt decomposition, while for the Birkhoff-Lewis case a suitable normal form theory for PDEs is also needed. For every result a statement is presented, with the main ideas of the proof, and with suggestions for further developments