In this book the authors determine the maximal subgroups of all the finite classical groups of dimension 12 or less. This work fills a long-standing gap in the literature. Behind this gap there is a story which I am pleased to have the opportunity to tell.

The completion of the classification of finite simple groups was first announced in the early 1980s. It was clear then (and before) that for many applications of the classification one would need detailed knowledge of the maximal subgroups of the simple groups and of their automorphism groups. Around that time, I gave a Part III course at Cambridge about the classification and its impact. Full of enthusiasm, I set a fearsome exam – I remember giving it to John Conway to check, and him saying that he couldn't do any of the questions, but he thought it was probably OK. The second highest mark was 18%, scored by a rather strong student. The top mark was 97%, scored by Peter Kleidman, a young American.

Soon afterwards, Kleidman started as my first research student. Michael Aschbacher had just published his fundamental theorem on maximal subgroups of the finite classical groups. The time seemed right to attempt to use this to determine all the maximal subgroups of the classical groups of low dimensions (up to 20, say, I thought optimistically). This was Kleidman's initial project.