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Compressible vortex sheets are fundamental waves,along with shock and rarefaction waves,in entropy solutions to multidimensional hyperbolic systems of conservation laws.Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions.For the Euler equations in two-dimensional gas dynamics,the classical linearized stability analysis oncompressible vortex sheets predicts stability when the Mach number M > (√)2 and instability when M < (√)2; and Artola-Majdas analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when M > (√)2.For the Euler equations in three-dimensions,every compressible vortex sheet is violently unstable and this violent instability is the analogue of the Kelvin-Helmholtz instability for incompressible fluids.In this talk,we will describe our recent research program on the mathematical analysis of compressible vortex sheets and related free boundary problems in fluid dynamics and magnetohydrodynamics (MHD).