It is well known that the motion of an incompressible fluid on a Riemannian manifold can be viewed as a geodesic flow for a certain right-invariant metric on on the group of volume-preserving diffeomorphisms. In my talk, I will review this framework and show how it can be extended to incorporate certain discontionus fluid motions, known as vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of "discontinuous diffeomorphisms". This is joint work with B.Khesin.