This paper studies \(n\)-exact functors of spaces and analyzes generalizations of two classical spectral sequences for such functors. The author defines an \(n\)-exact functor to be an endofunctor of the category of finite pointed \(CW\)-complexes and basepoint-preserving cellular maps that takes strongly co-Cartesian \((n+1)\)-cubical diagrams of spaces to co-Cartesian diagrams. Such functors are analogous to Goodwillie's \(n\)-excisive functors that transform strongly co-Cartesian \((n+1)\)-cubical diagrams into Cartesian diagrams, cf. \textit{T. G. Goodwillie} [K-Theory 5 (4), 295-332 (1992; Zbl 0776.55008)]. For an \(n\)-exact functor \(F\), the author continues the study of the generalized Atiyah-Hirzebruch spectral sequence (GAHSS) started in \textit{M. Hakim-Hashemi} and \textit{D. W. Kahn}, [ibid. 11(3), 241-257 (1997; Zbl 0891.55026)]. For a pointed \(CW\)-complex \(X\) and generalized homology theory \(E_*\), this spectral sequences converges to \(E_*(FX)\). In the case where \(F\) is a \(1\)-exact functor, Hakim-Hashemi and Kahn showed that \(E_*\circ F\) is a generalized homology theory and that the GAHSS for \(F\) and \(E_*\) is the classical Atiyah-Hirzebruch spectral sequence for \(E_*\circ F\). In the paper under review, the author analyzes the GAHSS in the case where \(F\) is a \(2\)-exact functor and \(X\) is a Moore space. He is able to identify the \(E^2\)-term of the GAHSS in terms of Baues' quadratic tensor product [\textit{H.-J. Baues}, Homotopy type and homology. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, (New York), Oxford Science Publications, (1996; Zbl 0857.55001) and Trans. Am. Math. Soc. 351 (2), 429--475 (1999; Zbl 0924.55007)]. He discusses the difficulties in carrying this work further to analyze the GAHSS for arbitrary \(n\)-exact functors. He also constructs a generalization of a spectral sequence of \textit{M. G. Barratt} [Colloq. algebr. Topology, Aarhus 1962, 22--27 (1962; Zbl 0142.21502)] and \textit{P. G. Goerss} [Q. J. Math., Oxf. II. Ser. 44 (173), 43--85 (1993; Zbl 0815.55005)] that he refers to as the Homological Barratt-Goerss spectral sequence. He uses this spectral sequence to prove several results about the behavior of \(H_*F(X)\) when \(F\) is an \(n\)-exact functor and \(X\) is a pointed \(CW\)-complex.