We all know what the differential of a smooth map from R to R is.
By looking at coordinates and then at charts, we also know what it is the differential of a smooth map between differentiable manifolds.
With a little bit of work, we can also define a (weak) differential for Sobolev maps in this setting (but the case of manifold-valued maps presents challenges already at this level).
In this talk I will discuss how it is possible to differentiate maps between spaces that have no underlying differentiable structure at all.
Applications to the study of spaces with Ricci curvature bounded from below will be discussed.