  We show that under some assumptions on the monodromy group some combinations
of higher Chern classes of flat vector bundles are torsion in the Chow group.
Similar results hold for flat vector bundles that deform to such flat vector
bundles (also in case of quasi-projective varieties). The results are motivated
by Bloch's conjecture on Chern classes of flat vector bundles on smooth complex
projective varities but in some cases they give a more precise information.
  We also study Higgs version of Bloch's conjecture and analogous problems in
the positive characteristic case.
