Vortrag: A report on regularity of minimal submanifolds Abstract: The area functional on submanifolds of a Riemannian manifold is one of the extnesively studied functionals in geometry. Its critical points, called minimal submanifolds, are the higher dimensional analogue of geodesics. In general dimensions, minimal submanifolds typically carry a small set of singularities. It has been a major challenge in Geometric Analysis to understand the nature of such singularities. This talk will first survey parts of the long and rich istory of this subject, and then present some recent progress in the theory in codimension 1, paying particular attention to branch point singularities.