The paper investigates some recent developments in the study of the motion of hypersurfaces in Riemannian manifolds when the normal speed is a monotonic function of the principal curvatures such that the evolution equation is a nonlinear parabolic PDE. In particular, the focus is on the mean curvature flow and, respectively, the inverse mean curvature flow. The author announces a new result, stemming from joint work with C. Sinestrari, on the singularities of the mean curvature flow in the larger class of hypersurfaces in \(\mathbb{R}^{n+1}\) with non-negative mean curvature. They derive estimates from below for all the elementary symmetric functions of principal curvatures to conclude that any rescaled limit of a singularity is (weakly) convex. This, together with older results of the author and R. Hamilton, respectively, leads to a description of all possible singularities in this mean convex case. The last section discusses the motion of surfaces by the inverse mean curvature, stemming from joint work with T. Ilmanen, and its applications to general relativity, in particular, the Riemannian Penrose inequality.