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We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time ``splash'' singularity, wherein the evolving 2-D hypersurface (the moving boundary of the fluid) self-intersects at a point. This requires solving the Euler equations on irregular domains whose boundaries form cusps. Our approach is founded upon the Lagrangian description of the free-boundary problem, combined with a novel approximation scheme of a finite collection of local coordinate charts; as such we are able to analyze a rather general set of geometries for the evolving 2-D free-surface of the fluid. A type of surgery is performed in the local chart covering the point of self-intersection. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems covering compressible flows and plasmas. This is joint work with D. Coutand.