Fluid flows in the presence of free surfaces occur in a great many situations in nature; examples include waves on the ocean and the flow of groundwater. I will discuss two kinds of existence results for free-surface problems: existence of solutions for the initial value problem, and existence of traveling waves. Progress on both the analysis and computing for these problems has been made possible by the introduction of geometrically motivated formulations. The well-posedness theorems for initial value problems are for three particular free surface problems: vortex sheets with surface tension, water waves, and interfacial flows in porous media. Each of these problems is considered in both two and three space dimensions. For the traveling wave problem, I will give a new formulation which allows for overturning waves, i.e., waves with multi-valued height. I will discuss an existence theorem for traveling vortex sheets with surface tension which uses this formulation, and I will show computational results for these waves. This includes joint work with Benjamin Akers, Nader Masmoudi, and J. Douglas Wright.