I will present an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that will be called of Tame Modulus. The normal form can be used to deduce informations on the dynamics of the system. In particular, in the nonresonant case, one gets that any small amplitude solution remains very close to a torus for very long times. Moreover, one gets a long time estimate of higher Sobolev norms of the solution. The theorem applies to several concrete examples ranging from the nonlinear wave equation with Dirichlet or periodic boundary conditions in one space dimensions to some particular NLS and plate equations in d–space dimensions. Several tools to deal with applications will be presented in the talk by Benoit Grebert.