Geometric flows have proved to be an indispensable tool in the analysis of a number of important problems in differential geometry and related fields. Particularly spectacular examples are the proofs of the Poincaré and geometrization conjectures using Ricci flow and the proof of the Riemannian Penrose inequality in general relativity using inverse mean curvature flow.

Further geometric applications include proofs of the Lusternik-Schnirelmann theorem and the Smale theorem (on Diff(S2)) using curve shortening flow, "local to global" results involving curvature using various intrinsic and extrinsic flows, and "pinched implies compact" theorems using Ricci flow.

There remain many interesting settings in which geometric flows may yield powerful new results.