The lecture shall trace the history of the theoretical study of the formation and evolution of shocks in compressible fluids, starting with the fundamental work of Riemann, the first work on nonlinear hyperbolic partial differential equations. Riemann considered the case of plane symmetry where the problem reduces to 1 spatial dimension. One milestone in the development of the theory was the work of Sideris who gave the first general proof of the finite time breakdown of smooth solutions in 3 spatial dimensions. Another milestone was the work of Majda who first addressed the problem of the local in time continuation of a shock front as a nonlinear free boundary problem for a nonlinear hyperbolic system of partial differential equations. I shall then discuss my own work, which uses differential geometric methods and resolves the resulting singularities giving a complete description in terms of smooth functions. My first work studies the maximal smooth development of given smooth initial data, the boundary of the domain of this development, and the behavior of the solution at this boundary. The boundary contains certain singular hypersurfaces which originate from certain singular surfaces. The singular surfaces do occur in nature, but not the singular hypersurfaces. My second work studies the physical evolution beyond the singular surfaces by solving a nonlinear free boundary problem with singular initial conditions associated to each of the singular surfaces. From each singular surface a shock hypersurface issues which appears as the corresponding free boundary. Speaker(s): Demetrios Christodoulou (ETH)