The study of crystallographic groups was originally related to one of the earliest applications of group theory to the natural sciences. These groups get their names because in three dimensions they occur as the symmetry groups of a crystal. In 1890, E. S. Fedorar classified these low-dimensional crystallographic groups, then in 1910 L. Bierberbach proved three theorems that were crucial to the understanding of crystallographic groups in arbitrary dimension. One could summarize these theorems the following way: the fundamental group of a flat manifold is a Bieberbach group, two flat manifolds with the same fundamental groups are homeomorphic, and up to equivalence there are only finitely many flat manifolds in each dimension. This set of theorems gave an affirmative answer to Hilbert’s eighteenth problem that there are only finitely many different space groups in n-dimensional Euclidean space.

*Geometry of Crystallographic Groups,* by Andrzej Szczepański, gives both an overview of the basic theory of crystallographic groups followed be a discussion of more advanced recent topics. The book concludes with a list of open questions to inspire the reader to delve further into this topic.

In the first few chapters the author gives a reasonable overview of the required knowledge, including several examples with references on the necessary background material. This is not the type of review one would expect for a course, but definitely adequate for someone undertaking their own study in the field. The subject of crystallographic groups comes from the marriages of several different fields of mathematics: differential topology, algebraic number theory, Riemannian geometry, cohomology of groups, and integral representations. As an individual who came to this text with a strong background in only one of these fields I found that the background material was reasonable and the author’s references were extremely helpful. When a theorem was presented for which another proof is known it was often included in an appendix or was given as a reference. Even though this book includes exercises it does not seem to be intended as a text book for a course.

*Bierberbach Groups and Flat Manifolds*, by Leonard S. Charlap, contains much of the material covered in the first five sections. Charlap’s text was written specifically for a graduate student audience; I felt that it went into a bit more detail than Szczepaήski’s text on the Bieberbach Theorems. However, this fulfills Szczepański’s stated goal to present a text as short as possible. It is clear that *Geometry of Crystallographic Groups* delves much further into recent discoveries than Charlap’s text. Additionally it is a nice resource that gives the reader a snapshot of current developments in this field. Overall I felt that this book was an excellent resource for an advanced graduate student or someone interested in learning about the current work on crystallographic groups.

Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle, IL. Her training is in computational group theory, particularly in using geometric properties of groups to solve algebraic problems in group extensions. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.