A central prediction in the study of spin glasses is that for any local reversible dynamics, one expects an exponential time to relaxation in the spin glass phase. In this talk, we prove this prediction for a broad class of Ising spin and spherical spin glass models. We present a single frame work to prove these estimates that applies equally in the discrete and manifold settings by introducing the notion of ``free energy barriers’’. The existence of free energy barriers will imply the the spectral gap of the dynamics is exponentially small, and thus that the mixing is exponentially fast. We then present sufficient conditions which imply the existence of these barriers for a large class of mean field spin glass models using the notions of the ``replicon eigenvalue’’, the 2D Guerra—Talagrand bounds, and a quenched LDP for the overlap distribution. I will report on two recent joint works with. G. Ben Arous (NYU) and R. Gheissari (NYU).