  We introduce Dolbeault cohomology valued characteristic classes of Higgs
bundles over complex manifolds. Flat vector bundles have characteristic classes
lying in odd degree de Rham cohomology and a theorem of Reznikov says that
these must vanish in degrees three and higher over compact K\"ahler manifolds.
We provide a simple and independent proof of Reznikov's result and show that
our characteristic classes of Higgs bundles and the characteristic classes of
flat vector bundles are compatible via the nonabelian Hodge theorem.
