When a steady supersonic flow impinges onto a solid wedge whose angle is less than a critical angle, so called detachment angle, there are two possible configurations: the weak shock solution and the strong shock solution. It is widely conjectured that the weak shock solution is physically admissible since it is the one observed experimentally. This is called `Prandtl's conjecture'. In this talk, I address this longstanding open conjecture, and present recent analysis to establish the stability theorem for steady weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters up to the detachment angle for potential flow. This talk is based on joint work with Gui-Qiang G. Chen(Univ. of Oxford) and Mikhail Feldman(UW-Madison).