In this edition we made minor corrections kindly pointed out to us by some c- leagues, and we updated and expanded the bibliography. We have not included the developments of the theory of gradient ?ows occurred in the last three years, asgradient?ows inspaceswith Alexandrovcurvaturebounds [135] (see also[119]) and Fokker-Planck equations in in?nite-dimensional spaces [18], largely based on the ideas developed in the book. We also mention the long survey paper [17], more focussed on gradient ?ows in Euclidean spaces with respect to the quadratic Wasserstein distance, where the notion of Evolution Variational Inequality is d- cussed more in detail, and the monumental book of C. Villani [147], which will surely become the standard reference for the theory of Optimal Transport and its applications to geometry and PDE’s. Pisa and Pavia, January 2008 Introduction This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 devoted to gradient ?ows in the L -Wasserstein space of probability measures on 2 a separableHilbert space X endowedwith the WassersteinL metric (we consider p the L -Wasserstein distance, p? (1,?), as well).