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. In this talk, I will discuss my contributions to the understanding of the systems of nonlinear partial differential equations which model such phenomena. The most important step in these results is making a suitable formulation of the problem. Influenced by the computational work of Hou, Lowengrub, and Shelley, we formulate the problems in natural, geometric variables. I will discuss my proofs (most of which are joint with Nader Masmoudi) of existence of solutions to the initial value problems for vortex sheets and water waves. I will also discuss computational results, including work with Jon Wilkening on the computation of special solutions, especially time-periodic interfacial flows.