Shock waves are steep fronts formed often in the compressible fluids in which convection dominates diffusion. They are fundamental in nature, especially in high-speed fluid flows. In this talk, we will start with various shock reflection-diffraction phenomena, von Nuemann’s conjectures for the existence of regular reflection-diffraction configurations, and their fundamental scientific issues and theoretical roles in the mathematical theory of multidimensional hyperbolic conservation laws. Then we will describe how these physical problems of shock reflection-diffraction can be formulated as mathematical free boundary problems involving nonlinear partial differential equations of mixed elliptic-hyperbolic type. Finally, we will discuss some recent developments in solving these free boundary problems and von Neumann’s conjectures to understand shock reflection-diffraction phenomena. Further trends and open problems in this direction, as well as their connections with some fundamental problems in elasticity, differential geometry, relativity, and other areas, will also be addressed if time permits. This talk is based mainly on our recent joint work with M. Feldman.