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Abstract: The heat flow of harmonic maps from a smooth, compact Riemannian manifold without boundary, (M,g) into another smooth, compact Riemannian manifold without boundary (N,h) was first studied in the seminal work of Eells and Sampson when the target manifold (N,h) has non-positive curvature. R.Hamilton studied the case when M has a compact  smooth boundary (and some special cases when N has also a smooth, compact boundary).  Gromov-Schoen studied harmonic maps from M  into a singular Cat(0) space which was used to  understand the p-adic superrigidity of lattices in groups of rank one.  A key analytical property of such harmonic maps is the Lipschitz continuity, from which one derives  Bochner type estimates and vanishing theorems.  As for Eells-Sampson theorem, it is rather natural to study the associated gradient (heat) flow,  and it has been a long open problem to construct suitable weak solutions in the singular setting.  In this talk, I shall describe an elliptic approach (which goes back to De Giorgi and also T. Ilmanen in the 1990s) to this problem both in the smooth and the singular settings, i.e. when the target is CAT(0) space. I will explain how to get Lipschitz bounds in the space variables (hence a suitable solution of the flow) and how this new approach offers as well a new viewpoint on the old problem of mappings between smooth manifolds. This is joint work with FH Lin, A Segatti and C Wang.