This talk will review some of the key ideas that went into the initiation of the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry. It will itemize reasons why physicists and geometers studying general relativity need to explore an approach which includes Riemannian and Lorentzian manifolds of lower regularity. To facilitate collaboration, the talk will survey various approaches to this challenging problem, emphasizing the work of participants in our program.