Lecture notes on differential calculus on RCD spaces

These notes are intended to be an invitation to differential calculus on RCD spaces. We start by introducing the concept of an “L 2 -normed L ∞ -module” and show how it can be used to develop a first-order (Sobolev) differential calculus on general metric measure spaces. In the second part of the manuscript we see how, on spaces with Ricci curvature bounded from below, a second-order calculus can also be built: objects like the Hessian, covariant and exterior derivatives and Ricci curvature are all well defined and have many of the properties they have in the smooth category.