The fluid interface  ``splash'' singularity was introduced by Castro, C\'{o}rdoba,  Fefferman, Gancedo, \& G\'{o}mez-Serrano.  A splash singularity occurs when a fluid interface remains locally smooth but self-intersects in finite time.   In this talk, I will very briefly discuss how we construct splash singularities for the one-phase 3-D Euler and Navier-Stokes equations.  I will then discuss the problem of two-phase Euler flow.   Recently, Fefferman, Ionescu, and Lie have shown that a locally smooth vortex sheet cannot self-intersect in finite time.   I will explain our proof of this result, which is based on elementary arguments and some precise blow-up rates for the gradient of the velocity of the fluid through which the interfaces tries to self-intersect.   This is joint work with D. Coutand.