The lecture describes joint work with Simon Brendle on the motion of closed hyper surfaces in a Riemannian manifold, on which the sum of any two principal curvatures is positive at all points. We exhibit a fully non-linear, parabolic choice of deformation law that preserves this curvature condition and allows only neck-like singularities which can be overcome by finitely many surgeries. The lecture explains the analytical techniques involved and sharp geometric conclusions made possible by this flow.