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Grothendieck's motives, as described in [Dem, K12,Ma] are designed as a tool to understand the cohomology of smooth projective varieties and the algebraic cycles modulo homological and numerical equivalence on them. According to Beilinson and Deligne, Grothendieck's category of pure motives should embed in a bigger category of mixed motives that allows the treatment of arbitrary varieties and an understanding of the whole Chow group of cycles modulo rational equivalence, in fact, even of all algebraic .fiT-groups of the varieties.In this paper we review some of these ideas and discuss some consequences. In particular, we show how the vast conjectural framework set up by Beilinson leads to very explicit conjectures on the existence of certain filiations on Chow groups of smooth projective varieties. These filtrations would offer an understanding of several phenomena and counterexamples that for some time have led people to believe that the behaviour of the algebraic cycles is absolute chaos for codimension bigger than one.In § 1 we review some basic facts on Chow groups, correspondences, and cycle maps into cohomology theories. We recall a counterexample of Mumford implying that in general Chow groups are not representable and the Abel-Jacobi map has a huge kernel and some investigations of Bloch on this topic.In § §2 and 4 we state altogether four versions of Beilinson's conjectures on mixed motives and filtrations on Chow groups, increasing in generality and sophistication. The first one does not even mention mixed motives and proposes finite filtrations (X) Q that are uniquely determined by their behaviour under algebraic correspondences. The first step is homological equivalence, but the following steps differ very much from those considered classically. For example, algebraic equivalence does not appear, and the second step is something like the kernel 1991 Mathematics Subject Classification. Primary 14C15; Secondary 14A20, 14C25. This paper is in final form and no version of it will be submitted for publication elsewhere.

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