For any G-module A of the group G, we have the Eilenberg-MacLane cohomology groups \(H^*(G,A)\). In general, they are quite difficult to compute. When A is the G-trivial module \({\mathbb{Z}}\), \(H^*(G,{\mathbb{Z}})\) becomes a ring, it is possible to get some information on \(H^*(G,{\mathbb{Z}})\) through the complex representation theory of G. Namely, if \(\rho\) : \(G\to GL(n,{\mathbb{C}})\), then we can use classifying space functor and pull back the cohomology ring of \(BGL(n,{\mathbb{C}})^{top}\). This latter is known to be a polynomial ring generated by the Chern classes. The present paper is a survey of many of the results on this theme. The original theme (to the reviewer's knowledge) first appeared in a work of \textit{M. F. Atiyah} [Publ. Math., Inst. Hautes Etud. Sci. 9, 241-288 (1961; Zbl 0107.023)]. Atiyah's paper seems to have been left out of the reference section by accident. This may be caused by the fact that \(H^*(G,{\mathbb{Z}})\) has not contributed as much information to finite groups G as the representation ring R(G). In spite of this comment by the author, there are some results in this area, and further investigations would be desirable. The survey is very well done. The reviewer would like to add the comment that many interesting groups (e.g. the infinite general linear group) have very few nontrivial representations (there are groups without any). In such cases, the theme leads to nothing. In contrast, the homology group \(H_*(G,{\mathbb{Z}})\) may become more interesting. To be more precise, the divisible elements of \(H_*(G,{\mathbb{Z}})\) may be permanently lost when we pass to \(H^*(G,{\mathbb{Z}})\). Of course, as a trade-off, \(H_*(G,{\mathbb{Z}})\) usually does not enjoy a ring structure.