  We study certain 'weights' for triangulated categories endowed with
$t$-structures. Our results axiomatize and describe in detail the relations
between the Chow weight structure (introduced in a preceding paper), the
(conjectural) motivic t-structure for Voevodsky's motives, and the conjectural
weight filtration for them. This picture becomes non-conjectural when
restricted to the derived categories of 1-motives (over a smooth base) and of
Artin-Tate motives over number fields; other examples include Beilinson's
derived category of graded polarizable mixed Hodge complexes and Saito's
derived category of mixed Hodge modules. We also study weight filtrations for
the heart of $t$ and (the degeneration of) weight spectral sequences. In a
subsequent paper we apply the results obtained in order to prove that (certain)
'standard' conjectures imply the existence of Beilinson's categories of mixed
motivic sheaves endowed with weight filtrations.
