The main result of this talk is a global existence theorem for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description of the solution which shows that modified scattering holds. The main tools used in the proof are, on the one hand, a normal forms paradifferential method allowing one to obtain energy estimates on the Eulerian formulation of the water waves equation. On the other hand, we prove uniform bounds interpreting the equation in a semi-classical way, and combining Klainerman vector fields with the description of the solution in terms of semi-classical Lagrangian distributions.