  Shock waves are fundamental in nature. One of the most fundamental problems
in fluid mechanics is shock reflection-diffraction by wedges. The complexity of
reflection-diffraction configurations was first reported by Ernst Mach in 1878.
The problems remained dormant until the 1940s when John von Neumann, as well as
other mathematical/experimental scientists, began extensive research into all
aspects of shock reflection-diffraction phenomena. In this paper we start with
shock reflection-diffraction phenomena and historic perspectives, their
fundamental scientific issues and theoretical roles in the mathematical theory
of hyperbolic systems of conservation laws. Then we present how the global
shock reflection-diffraction problem can be formulated as a boundary value
problem in an unbounded domain for nonlinear conservation laws of mixed
hyperbolic-elliptic type, and describe the von Neumann conjectures: the sonic
conjecture and the detachment conjecture. Finally we discuss some recent
developments in solving the von Neumann conjectures and establishing a
mathematical theory of shock reflection-diffraction, including the existence,
regularity, and stability of global regular configurations of shock
reflection-diffraction by wedges.
