This paper concerns the deformation of hypersurfaces in Riemannian manifolds using fully nonlinear parabolic equations defined in terms of the Weingarten curvature. It is shown that any initial hypersurface satisfying a natural convexity condition produces a solution which converges to a single point in finite time, and becomes spherical as the limit is approached. The result has topological implications, including a new proof of the 1/4-pinching sphere theorem, and a new “dented sphere” theorem which allows some negative curvature. The equations considered are related to the flow by mean curvature, but do not include this. An example of an evolution equation which does satisfy the conditions is the flow by harmonic mean curvature, where the speed of the hypersurfaces is given by the harmonic mean of the principal curvatures at each point. The results obtained are significantly stronger than those known for the mean curvature.