This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in \(K(B)\). We also construct weight structures for Voevodsky's categories of motives and for various categories of spectra. A weight structure \(w\) defines Postnikov towers of objects; these towers are canonical and functorial 'up to morphisms that are zero on cohomology'. For \(Hw\) being the heart of \(w\) (in \(DM_{gm}\) we have \(Hw=Chow\)) we define a canonical conservative 'weakly exact' functor \(t\) from our \(C\) to a certain weak category of complexes \(K_w(Hw)\). For any (co)homological functor \(H:C\to A\) for an abelian \(A\) we construct a weight spectral sequence \(T:H(X^i[j])\implies H(X[i+j])\) where \((X^i)=t(X)\); it is canonical and functorial starting from \(E_2\). This spectral sequences specializes to the 'usual' (Deligne's) weight spectral sequences for 'classical' realizations of motives and to Atiyah-Hirzebruch spectral sequences for spectra. Under certain restrictions, we prove that \(K_0(C)\cong K_0(Hw)\) and \(K_0(End C)\cong K_0(End Hw)\). The definition of a weight structure is almost dual to those of a t-structure; yet several properties differ. One can often construct a certain \(t\)-structure which is 'adjacent' to \(w\) and vice versa. This is the case for the Voevodsky's \(DM^{eff}_-\) (one obtains certain new Chow weight and t-structures for it; the heart of the latter is 'dual' to \(Chow^{eff}\)) and for the stable homotopy category. The Chow t-structure is closely related to unramified cohomology.
