We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where this does not occur, non-uniformly convex regions and singular parts in the initial hypersurface
may persist, including flat sides, ridges of infinite curvature, or ‘cylindrical’ regions where some of the
principal curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in
such cases the hypersurface may even collapse to a line segment or higher-dimensional disc rather than
to a point. We provide sufficient conditions for these various disasters to occur, and by avoiding these
arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become
smooth and uniformly convex and contract to points.