  We consider classical curvature flows: 1-parameter families of convex
embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary
(non-homogeneous) function of the radii of curvature. The associated flow of
the radii of curvature is a second order system of partial differential
equations which we show decouples to highest order. We determine the conditions
for this system to be parabolic and investigate the lower order terms. The
zeroth order terms are shown to form a Hamiltonian system, which is therefore
completely integrable.
  We find conditions on parabolic curvature flows that ensure boundedness of
various geometric quantities and investigate some examples, including powers of
mean curvature flow, Gauss curvature flow and mean radius of curvature flow, as
well as the non-homogeneous Bloore flow.
  As a new tool we introduce the Radii of Curvature diagram of a surface and
its canonical hyperbolic metric. The relationship between these and the
properties of Weingarten surfaces are also discussed.
