  We relate the notion of finite dimensionality of the Chow motive M(X) of a
smooth projective variety X (as defined by S. Kimura) with the Conjectures of
Beilinson, Bloch and Murre on the existence of a filtration on the Chow ring
CH(X). We show (Theorem 14) that finite dimensionality of M(X) implies
uniqueness, up to isomorphism, of Murre's decomposition of M(X). Conversely
(Theorem 17), Murre's Conjecture for a suitable m-fold product of X by itself
implies finite dimensionality of M(X). We also show (Theorem 27) that, for a
surface X with trivial geometrical genus, the motive M(X) is finite dimensional
if and only if the Chow group of 0-cycles of X is finite dimensional in the
sense of Mumford, i.e. iff the Bloch Conjecture holds for X.
