Our main goal is to review recent progress on the long-time regularity of solutions of the water waves equations, in two and three dimensions.

Using the Eulerian formulation, we will present global existence results for gravity waves and gravity-capillary waves in 2D and 3D, in a unified framework. We will stress the importance of resonances and present a general method to control the solution and prove dispersion and decay.

We will also discuss some recent results in periodic domains, with the goal of understanding the long-term effective dynamics of the water waves system, and derive a wave kinetic equation.