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We are concerned with -convergence of gradient ows, which is a notion meant to ensure that if a family of energy functionals depending of a parameter -converges, then the solutions to the associated gradient ows converge as well. In this paper we present both a review of the abstract \theory" and of the applications it has had, and a generalization of the scheme to metric spaces which has not appeared elsewhere. We also mention open problems and perspectives. -convergence was introduced by De Giorgi in the 70’s. It provides a convenient notion of convergence of a family of energy functionalsE" to a limiting functionalF , which ensures in particular that minimizers ofE" converge to minimizers ofF . In [DeG1], De Giorgi raised the question of knowing whether there was any general relation between solutions of the

Gamma-convergence of gradient flows on Hilbert and metric spaces and applications