### Abstract

I shall present a sigma-model approach to the out of equilibrium dynamics of glassy systems. First, I shall review some salient features of the relaxation of glassy systems. Next, I shall describe the main idea on which the theoretical approach is based: the development of an approximate asymptotic time-reparametrization invariance. I shall briefly describe numeric and analytic studies of several models (finite dimensional spin-glasses, kinetically constrained models, elastic lines in random environments and the O(N) model for coarsening in the large N limit) that put to the test our predictions. I shall end by discussing the general picture that emerges. This work is the result of a collaboration with Claudio Chamon (Boston University) et al.