Vortex sheets provide a framework for drastically reducing the computational cost of simulating high-Reynolds-number fluid flows. Traditional grid-based solvers for the Navier-Stokes equations face computational bottlenecks in modelling systems such as aerodynamic wakes or thin shear layers due to the fine mesh resolution required to capture detailed structures. Vortex sheet modeling circumvents this by reducing the problem of determining the full two-dimensional fluid motion to tracking the evolution of a one-dimensional curve. In this talk, we will derive the equations of motion governing vortex sheet dynamics and discuss numerical and modelling challenges which arise. We will apply the vortex sheet model to simulate a number of different systems, including the evolution of a thin free shear layer and the wake shed by a swimming fish. Along the way, we will also get the chance to admire the striking beauty and complexity of vortex sheet motion!