Consider a Hamiltonian PDE having an elliptic equilibrium at zero. Assuming a suitable conditions on the eigenvalues of the linearized problem (frequncies of small oscillation) we will construct a canonical transformation putting the system in Birkhoff normal form up to a small reminder. In the nonresonant semilinear case one can deduce that solutions corresponding to small initial data remain close to approximatively invariant tori for long times. A similar conclusion can be obtained also in the quasilinear case provided some assumptions on the Lyapunov exponents of the system are added. The general theory will be applied to a quasilinear wave equations in an n dimensional paralleliped and to the equations of the water wave problem. An extension of the result to some systems with continuous spectrum will be discussed; as an application it will be shown that the nonlinear Schr¨odinger equation appears as a resonant normal form for the nonlinear Klein–Gordon equation. Some open problems will be discussed.