Recent advances on the global regularity for irrotational water waves

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach, as in [7, 1, 2]. We then describe in a unified framework, using the Eulerian formulation, global existence results for three dimensional and two dimensional gravity waves, see [70, 146, 145, 87, 5, 6, 79, 80, 136], and our joint result with Deng and Pausader [60] on global regularity for the 3D gravity-capillary model. We conclude this review with a short discussion about the formation of singularities, and give a few additional references to other interesting topics in the theory.