We start by introducing the free boundary problem for the Euler equations and discuss some aspects related to its local well-posedness, in both Eulerian and Lagrangian coordinates, together with some known results. We will then describe some general tools that have been used to construct global solutions for the irrotational problem with gravity in 3 space dimensions. In the second part we will focus on the gravity water waves system in 2 dimensions and sketch the proof of global existence of small solutions. This is based on energy estimates performed in some special Lagrangian coordinates due to S. Wu, and on decay estimates which are proven using the Eulerian formulation of the system.