When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. The complexity of reflection configurations was first reported by Ernst Mach in 1878, and experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most fundamental issues for shock reflection have not been understood, including the transition of the different patterns of shock reflection. Therefore, it becomes essential to establish a mathematical theory on the existence, stability, and regularity of global configurations of shock reflection, especially for potential flow which has widely been used in aerodynamics.

In this talk we will start with various shock reflection phenomena and their fundamental scientific issues. Then we will describe how the shock reflection problems can be formulated as free boundary problems for nonlinear conservation laws of mixed-composite hyperbolic-parabolic type. Finally we will discuss some recent developments in attacking the shock reflection problems, including the existence, stability, and regularity of global configurations of regular shock reflection by wedges for potential flow. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. This talk will be mainly based on the joint work with M. Feldman.