Abstract: In this talk, I will present recent results on the stability of vortex sheets in compressible flows. Vortex sheets are contact discontinuities of the underlying flows. The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis. The formulation of the vortex sheet problem will be introduced. In particular, I will explain the difficulties of the vortex sheets problem and the ideas of proofs will be discussed.