In 1981 the importance of homological algebra as a tool in group theory was beginning to be recognised. After the pioneering work in the 1940's by S. Eilenberg, S. MacLane and B. Eckmann on the homology and cohomology of groups, twenty years elapsed before really convincing applications appeared: the prime example was Gaschütz's famous theorem on the existence of outer automorphisms of finite p-groups. The well known sets of notes by K. W. Gruenberg and U. Stammbach, which were published in the 1970's, had proved to be a stimulus to research, and already a body of work had appeared in the literature. It seemed timely to write a survey for Groups St Andrews 1981.

The twenty five years which have elapsed since that critical conference have witnessed a continuation of the trend in group theory to introduce techniques from homological algebra, as well as other areas of mathematics. Today many group theorists are conversant with a variety of homological methods, including spectral sequences. Our aim here is to survey some of the achievements during this period.

Until about 1980 group theoretic interpretations of the cohomology groups Hn(G,M) had only been found only for n ≤ 3; these arise of course from the classical theory of group extensions. The problem of finding group theoretic interpretations of Hn(G,M) for arbitrary n was solved by D. F. Holt and J. Huebschmann.