We study enhancement of diﬀusive mixing on a compact Riemannian manifold by a fast incompressible ﬂow. Our main result is a sharp description of the class of ﬂows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the ﬂow amplitude is large enough. The necessary and suﬃcient condition on such ﬂows is expressed naturally in terms of the spectral properties of the dynamical system associated with the ﬂow. In particular, we ﬁnd that weakly mixing ﬂows always enhance dissipation in this sense. .