This article concerns the description of the Hochschild and cyclic homologies for algebras \(S=\bigoplus_{g\in G}S_g\) that are graded by a group \(G\). Some special cases have been treated earlier, notably group algebras (by Burghelea), skew group rings (by Feigin and Tsygan), and strongly \(G\)-graded algebras (Hochschild homology only, by the reviewer). For finite groups \(G\), the study of \(G\)-graded algebras can be reduced, in some sense, to the case of skew group rings by using Cohen-Montgomery duality in conjunction with Morita invariance of the homology theories in question. For infinite \(G\), a similar approach is successful, based on D. Quinn’s extension of Cohen-Montgomery duality. This leads to the main results of the article which express the homology of \(S\) in terms of (hyper-) homology of \(G\) with suitable coefficients. More precisely, it is easy to see that the cyclic homology \(HC_*(S)\) of \(S\) has a decomposition \(HC_*(S)=\bigoplus_{[g]}HC_*(S)_{[g]}\), where \([g]\) ranges over the conjugacy classes of elements \(g\in G\), and similarly for Hochschild homology \(H_*(S)\). Thus the issue is to describe the components \(HC_*(S)_{[g]}\) and \(H_*(S)_{[g]}\). The description obtained in the article has an especially simple form in the case where the algebra \(S\) is strongly \(G\)-graded. In this case, the components are accessible as limits of suitable spectral sequences. For example, if \(g\in G\) has infinite order, then there is a spectral sequence \((E^2_{p,q})_{[g]}=H_p(C_G(g)/\langle g\rangle,H_q(S_1,S_g))\Rightarrow HC_n(S)_{[g]}\). Here, \(C_G(g)\) is the centralizer of \(g\) in \(G\) and \(H_q(S_1,S_g)\) is the Hochschild homology of \(S_1\) with coefficients in the bimodule \(S_g\). The last part of the article gives an application of this material to the Bass conjecture on the structure of idempotents or, more generally, the support of the Hattori-Stallings rank function for finitely generated projective modules over group algebras. Some results of Eckmann in this direction is extended.