  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-diffraction configurations was first reported by Ernst Mach in 1878,
and experimental, computational, and asymptotic analysis has shown that various
patterns of shock reflection-diffraction configurations may occur, including
regular reflection and Mach reflection. In this paper we start with various
shock reflection-diffraction phenomena, their fundamental scientific issues,
and their theoretical roles as building blocks and asymptotic attractors of
general solutions in the mathematical theory of multidimensional hyperbolic
systems of conservation laws. Then we describe how the global problem of shock
reflection-diffraction by a wedge can be formulated as a free boundary problem
for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type.
Finally we discuss some recent developments in attacking the shock
reflection-diffraction problem, including the existence, stability, and
regularity of global regular reflection-diffraction solutions. 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. Further trends and open problems in this
direction are also addressed.
