We consider the evolution of compact surfaces by fully nonlinear, parabolic curvature
ows for which the normal speed is given by a smooth, degree one homogeneous function of the principal curvatures of the evolving surface. Under no further restrictions on the speed function, we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow. This generalises the corresponding result  of Huisken
and Sinestrari for the mean curvature ow to the largest possible class of degree one homogeneous surface flows.