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

Weights and t-structures: in general triangulated categories, for 1-motives, mixed motives, and for mixed Hodge complexes and modules