For transformation group G of a topological space X a spectral sequence with the term E p, q 2 = H p ( G; H q ( X; A)), where A is a coefficient group, is introduced. The cohomology of th e complex of G-invariant cochains with coefficients in A determine the characteristic classes of G as elements of E ∞; in particular cases they are elements of H p ( G; H q ( X; A)) ( or H p c ( G; H q ( X; A)) for continous transformation group G). The main applications concern the case when A = R and the de Rham complex is used. It is shown that the characteristic classes of Reiman, Semenov- Tian-Shansky and Fadeev for the automorphism group of a smooth principal fibre bundle and characteristic classes of Bott for the diffeomorphism group of a manifold are the partial cases of the above construction. The connection of the above characteristic classes with the functional of Atiyah-Patodi-Singer is indicated.