These notes are based on lectures that I have given at various times during the last four years and at various places, but mainly at Queen Mary College, London. Chapters 1 to 7 have been in circulation as a volume in the Queen Mary College Mathematics Notes since the autumn of 1967. They are reproduced here unchanged except for the addition of some bibliographical material and the correction of some minor errors.
Chapter 8 is an attempt at a reasonably complete survey of the subject of finite cohomological dimension. I have included proofs of everything that is not readily accessible in the literature.
Chapters 9 and 11 contain an account of a kind of globalised extension theory which I believe to be new. A survey of some of the results has appeared in volume 2 of "Category theory, homology theory and their applications", Springer lecture Notes, no.92 (1969). The basic machinery of extension categories for arbitrary groups is given in chapter 9. Then in chapter 11 we focus attention exclusively on finite groups and primarily on the structure of minimal projective extensions. Chapter 10 is purely auxiliary and merely sets out some cohomological facts needed in chapter 11.

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