Explore the concept of differentiation in non-differentiable environments through this 43-minute conference talk delivered by Nicola Gigli at the Workshop on "Non-regular Spacetime Geometry" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the challenges of defining differentials beyond traditional smooth maps between differentiable manifolds. Discover how to differentiate maps between spaces lacking underlying differentiable structures. Examine the applications of these concepts to the study of spaces with Ricci curvature bounded from below. Gain insights into advanced mathematical techniques that push the boundaries of conventional differential calculus and their potential implications for non-regular spacetime geometry.