On spectral sequences for Postnikov towers

In this paper we construct a natural spectral sequence corresponding to a Postnikov tower (system) along with a (co)homological functor from a triangulated category. We apply the results to some classical filtrations in the homotopy category $K(A)$. In triangulated categories endowed with weight structures, we constuct weight spectral sequences (as defined by M. V. Bondarko), with some additional properties.