Each real analytic symplectic diffeomorphism admitting a non-resonant fixed point can be formally conjugated to a formal integrable symplectic diffeomorphism, its Birkhoff Normal Form (BNF). I shall discuss two questions. The first one is due to H. Eliasson: are there examples of divergent BNF? The second is the following: is it possible to perturb in the real analytic topology a given real analytic symplectic diffeomorphism so that its BNF converges?