Abstract: The motion of interfaces in fluids is central to a remarkable number of natural and engineered systems: some examples are the swimming of fish, waves on the ocean, and the flow of blood. Fluid interfaces can exert geometric forces on the surrounding fluid (e.g. surface tension) and can also have inertial forces (e.g., a heavy membrane). Consequently, the equations describing their motion are both highly nonlinear and nonlocal, making both analytical and numerical studies difficult. In this talk, I will discuss my solution of one such analytical problem, the well-posedness of vortex sheets with surface tension; this work is heavily influenced by sophisticated numerical methods developed to evolve such interfaces. I will discuss an extension of these ideas to both the analysis and simulation of "inertial interfaces," and discuss how new analysis for the 3D vortex sheet problem may be suggesting new routes to numerical simulation.