The theory of metric measure spaces with upper dimension and lower Ricci curvature bounds has provided an extremely fruitful approach to nonsmooth Riemannian geometry in recent years. In this talk we explore an analogous framework for nonsmooth Lorentzian geometry, based on entropic convexity conditions along timelike geodesics of probability measures on spacetime. Based in part on work with Clemens Sämann (University of Vienna) [68][79] at http://www.math.toronto.edu/mccann/publications