Differential calculus on metric measure spaces

In this talk I will describe in which sense general metric measure spaces possess a first order differential structure. In particular we shall see how to define the differential of sufficiently regular (in fact Sobolev) functions. If time permits I will say few words about spaces with Ricci curvature bounded from below: in this setting also a second order calculus is possible, with objects like Hessian, covariant/exterior derivative and Ricci curvature which are all well defined.