This book is essentially a text book that introduces the geometrical objects which arise in the study of vector spaces over finite fields. It advances rapidly through the basic material, enabling the reader to consider the more interesting aspects of the subject without having to labour excessively. There are over a hundred exercises which contain a lot of content not included in the text. This should be taken into consideration and even though one may not wish to try to solve the exercises themselves, they should not be ignored. There are detailed solutions provided to all the exercises.

The first four chapters treat the algebraic and geometric aspects of finite vector spaces. The following three chapters consist of combinatorial applications. There is a chapter containing a brief treatment of applications to groups, real geometry, codes, graphs, designs and permutation polynomials. Then there is a chapter that gives a more in-depth treatment of applications to extremal graph theory, specifically the forbidden subgraph problem, and then a chapter on maximum distance separable codes.

This book is self-contained in the sense that any theorem or lemma which is subsequently used is proven. The only exceptions to this are Bombieri's theorem and the Huxely–Iwaniec theorem concerning the distribution of primes, which are used in the chapter on the forbidden subgraph problem, the Hasse– Weil theorem, which is used to bound the number of points on a plane algebraic curve at the end of the chapter on maximum distance separable codes, and Hilbert's Nullstellensatz, which is used in the appendix on commutative algebra. Although there are almost no prerequisites, it would be helpful to have studied previously some basic algebra and linear algebra, since otherwise the first couple of chapters may appear somewhat brief. There are some theorems that are quoted without proof, but in all cases these appear at the end of some branch and are not built upon.