  In this paper we investigate Murre's conjecture on the Chow--K\"unneth
decomposition for two classes of examples. We look at the universal families of
smooth curves over spaces which dominate the moduli space $\cM_g$, in genus at
most 8 and show existence of a Chow--K\"unneth decomposition. The second class
of examples include the representation varieties of a finitely generated group
with one relation. This is done in the setting of equivariant cohomology and
equivariant Chow groups to get equivariant Chow--K\"unneth decompositions.
