  A classical topic in the mathematical theory of hydrodynamics is to study the
evolution of the free surface separating air from an incompressible perfect
fluid. The goal of this survey is to examine this problem for two important
sets of equations: the water wave equations and the Hele-Shaw equations,
including the Muskat problem. These equations are different in nature,
dispersive or parabolic, but we will see that they can be studied using related
tools. In particular, we will discuss a paradifferential approach to these
problems.
