abstract: The harmonic map flow of Eells and Sampson is a gradient flow of the harmonic map energy of a map between Riemannian manifolds, which tries to flow to harmonic maps. The critical case is when the domain is two-dimensional. I will describe another gradient flow on the same energy, which is adapted to find minimal surfaces. I will survey what has been done over the last few years, and sketch some forthcoming work on the large time asymptotics. Joint work with Melanie Rupflin and others.