The question of vacuum formation in compressible flows is a long-standing open in the mathematical theory of fluid dynamics. Roughly stated the issue is: do there exist solutions of the Navier-Stokes system for viscous compressible flow that exhibit vacuum (vanishing density) in finite time, when the initial density density is bounded away from zero? The problem is relevant from a modeling perspective as well as for theoretical reasons. The question has been answered negatively in the one-dimensional case (Hoff & Smoller), 2001; the two and three dimensional cases are open. A computational study of the above issues indicates vacuum formation. These results, together with the pertaining numerical difficulties, will be discussed. The important limit case (infinite Reynolds number) of the Euler equations of gas dynamics, for which vacuum formation is well known, will also be recalled. Finally, ongoing work on the numerical study of regularity losses of solutions to the relativistic Euler equations will be briefly discussed. Joint work with Kristen DeVault (North Carolina State University), Kris Jenssen (PennState) and Joel Smoller (U. of Michigan).