Explore the intricacies of curvature tensors and measures in singular spaces through this 50-minute lecture delivered by Andreas Bernig at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into advanced geometric concepts as part of the Thematic Programme on "Geometry beyond Riemann: Curvature and Rigidity." Gain insights into the mathematical foundations that extend beyond traditional Riemannian geometry, focusing on the analysis of singular spaces and their unique curvature properties.