abstract: We describe a procedure to define mean curvature flow of hypersurfaces after singularities based on a surgery procedure. The construction is inspired by the work of Hamilton for the Ricci flow; compared with the usual notions of weak solutions, it has the advantage that it keeps track of the changes in the topology of the surface and is therefore more suitable for geometric applications. We describe the main steps of our analysis and compare our construction with the recent results of Perelman. The work is in collaboration with G. Huisken (Max-Planck-Insitut, Potsdam).