  We study the ring of characteristic classes with values in the Chow ring for
principal $G$-bundles over schemes. If we consider bundles which are locally
trivial in the Zariski topology, then we show, for $G$ reductive, that this
ring is isomorphic to the Weyl group invariants in the algebra generated by
characters of the maximal torus. For general principal bundles the same
isomorphism holds after tensoring the coefficients with ${\Bbb Q}$. As a
corollary, we show that any (non-torsion) topological characteristic class is
algebraic when applied to Zariski locally trivial bundles over complex
algebraic varieties.
