  The Bloch-Beilinson-Murre conjectures predict the existence of a descending
filtration on Chow groups of smooth projective varieties which is functorial
with respect to the action of correspondences and whose graded parts depend
solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish
to explore, at the cost of having to assume general conjectures about algebraic
cycles, how the coniveau filtration on the cohomology of $X$ has an incidence
on the Chow groups of $X$. However, by keeping such assumptions minimal, we are
able to prove some of these conjectures either in low-dimensional cases or when
a variety is known to have small Chow groups. For instance, we give a new
example of a fourfold of general type with trivial Chow group of zero-cycles
and we prove Murre's conjectures for threefolds dominated by a product of
curves, for threefolds rationally dominated by the product of three curves, for
rationally connected fourfolds and for complete intersections of low degree.
The BBM conjectures are closely related to Kimura-O'Sullivan's notion of
finite-dimensionality. Assuming the standard conjectures on algebraic cycles
the former is known to imply the latter. We show that the missing ingredient
for finite-dimensionality to imply the BBM conjectures is the coincidence of a
certain niveau filtration with the coniveau filtration on Chow groups.
