  The main goal of this paper is to define the so-called Chow weight structure
for the category of Beilinson motives over any 'reasonable' base scheme $S$
(this is the version of Voevodsky's motives over $S$ defined by Cisinski and
Deglise). We also study the functoriality properties of the Chow weight
structure (they are very similar to the well-known functoriality of weights for
mixed complexes of sheaves). As shown in a preceding paper, the Chow weight
structure automatically yields an exact conservative weight complex functor
(with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of the Chow weight
structure; it is 'generated' by motives of regular schemes that are projective
over $S$. Besides, Grothendiek's group of $S$-motives is isomorphic to
$K_0(Chow(S))$; we also define a certain 'motivic Euler characteristic' for
$S$-schemes. We obtain (Chow)-weight spectral sequences and filtrations for any
cohomology of motives; we discuss their relation to Beilinson's 'integral part'
of motivic cohomology and to weights of mixed complexes of sheaves. For the
study of the latter we introduce a new formalism of relative weight structures.
