We’ll consider the dynamics of localized vortex patches in the model of the Euler equations for 2-D incompressible, inviscid fluid flows. The solutions of the Euler equations describing such motion, as well as the flow maps they generate, have limited regularity but nevertheless are uniquely defined for all times [V.I. Youdovich ’63]. Moreover, the boundaries of initially smooth vortex patches retain their structure [J.-Y. Chemin ’93]. Results of numerical simulation have been reported that show that a corner singularity in the boundary of a vortex patch evolves into a cusp [A. Cohen, R. Danchin ’00]. We'll give an analytical proof of this observation and show, in fact, that the cusp forms instantaneously. This phenomenon can also be traced in the motion of compressible flows, modeled by solutions of the Navier-Stokes equations. This is joint work with David Hoff (Indiana University).