  This is an expository paper on the theory of gradient flows, and in
particular of those PDEs which can be interpreted as gradient flows for the
Wasserstein metric on the space of probability measures (a distance induced by
optimal transport). The starting point is the Euclidean theory, and then its
generalization to metric spaces, according to the work of Ambrosio, Gigli and
Savar{\'e}. Then comes an independent exposition of the Wasserstein theory,
with a short introduction to the optimal transport tools that are needed and to
the notion of geodesic convexity, followed by a precise desciption of the
Jordan-Kinderleher-Otto scheme, with proof of convergence in the easiest case:
the linear Fokker-Planck equation. A discussion of other gradient flows PDEs
and of numerical methods based on these ideas is also provided. The paper ends
with a new, theoretical, development, due to Ambrosio, Gigli, Savar{\'e},
Kuwada and Ohta: the study of the heat flow in metric measure spaces.
