The problem of existence and qualitative behavior of solutions of evolution equations is a classical one in the theory of PDEs. In this colloquium I will focus on the use of Birkhoff normal form for the proof of the so called almost global existence results in Hamiltonian PDEs. Such results deal with perturbation of linear hyperbolic equation (for example the wave equation) and ensure that solutions corresponding to smooth and small initial data remain small and smooth for times of order $\epsilon^{-r}$, $\forall r$. Here $\epsilon$ is the size of the initial datum.

Nowadays there exists a well established theory for semilinear equations in space dimension 1 and recently also the situation of quasilinear equations still in dimension 1 has been understood.

During the last years some results have been obtained also for higher dimensional domains, but only for semilinear equations.

In this colloquium I will review the classical theory presenting the main ideas and then I will present some of the most recent results trying to put into evidence the main tools that have been developed in order to deal with the problem.