Let \(k\) be an arbitrary field. Then an algebraic \(k\)-scheme is by definition a scheme of finity type over \(k\), and an algebraic group \(G\) over \(k\) is just a group object in the category of algebraic schemes and their morphisms. Classically, algebraic groups were group objects in the category of algebraic varieties defined over \(k\), and as such they have always been of special significance in modern algebraic geometry, playing a similar role as Lie groups play in differential geometry and analysis. Especially the closed algebraic subgroups of GL\(_n\), the so-called linear algebraic groups, are of central importance in various areas of algebraic geometry, particularly in the construction of classifying spaces (or moduli spaces). Despite the prominence of algebraic groups in contemporary algebraic geometry, an adequate systematic, comprehensive and up-to date textbook on the subject did not exist in the last fifty years. Actually, the standard references by J. E. Humphreys [Linear algebraic groups. Springer, New York (1975; Zbl 0325.20039)], A. Borel [Linear algebraic groups. 2nd enlarged ed. New York etc.: Springer-Verlag (1991; Zbl 0726.20030)], or T. A. Springer [Encycl. Math. Sci. 55, 1–121 (1994; Zbl 0789.20044); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 55, 5–136 (1989)], all titled “Linear algebraic groups”, are by now serving for decades, and are thematically as specialized as their common title indicates. As for the more general theory of group schemes, the only (modest) introductions even date back to 1964 [M. Demazure (ed.) and A. Grothendieck (ed.), Séminaire de géométrie algébrique du Bois Marie 1962-64. Schémas en groupes (SGA 3). Tome I: Propriétés générales des schémas en groupes. New annotated edition of the 1970 original published bei Springer. Paris: Société Mathématique de France (2011; Zbl 1241.14002)] and 1970 [M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique. Généralités. Groupes commutatifs. Avec un appendice ‘Corps de classes local’ par Michiel Hazewinkel. (1970; Zbl 0203.23401)].
However, now there is the voluminous new textbook under review, written by the well known researcher and author of numerous textbooks, monographs, and lecture notes, James S. Milne. As the author points out in the preface, this book represents his attempt to write such an overdue, modern successor to the above-mentioned earlier standard texts, thereby providing an exposition of the general theory of group schemes of finite type over a field, in the language of modern algebraic geometry, but assuming its very basics only.
The result, namely the present book, may be seen as the most systematic, comprehensive, and actual treatise on the subject of today.
The text comprises twenty-five chapters, each of which is subdivided into several (mostly quite short) sections. Roughly, there are five main parts arranged as follows: 5mm

A.

Basic theory of general algebraic groups (Chapters 1–8).
This part presents the general theory of algebraic group schemes over a field, including their basic properties, algebraic subgroups, kernels, group actions, transporters, normalizers, and centralizers. Furthermore, examples and basic constructions affine algebraic groups and Hopf algebras, linear representations of algebraic groups, isomorphism theorems, solvable and nilpotent algebraic groups, actions of algebraic groups on schemes, flag varieties, and the structure theory of general algebraic groups are discussed.

B.

Preliminaries on affine algebraic groups (Chapters 9–11).
These three chapters serve as a preparation for the more detailed study of affine algebraic group schemes later on. They treat basic Tannakian theory, that is the relation between an algebraic group and its category of representations, the Lie algebra of an algebraic group, and the special case of finite group schemes.

C.

Solvable affine algebraic groups (Chapters 12–16).
The five chapters forming this part study solvable group schemes via their representations. This includes diagonalizable groups, groups of multiplicative type, linearly reductive groups, tori, torus actions on schemes, Luna maps, the Białynicki-Birula decomposition, unipotent algebraic groups, Hochschild cohomology and Hochschild extensions, and the general structure theory for algebraic groups. In the course of the latter, trigonalizable algebraic groups, commutative algebraic groups, nilpotent algebraic groups, and split algebraic groups are characterized as well.

D.

Reductive algebraic groups (Chapters 17–25).
This is the main part of the book. In Chapters 17 to 23, the author develops in great detail the general structure theory of split reductive groups and their representations. Based on their root data, this leads to the description of all the almost-simple algebraic groups in Chapter 24, and the last Chapter 25 treats some additional topics, mainly a sketch of how the theory of split groups can be extended to non-split groups.
More precisely, the nine chapters of Part D are devoted to the following topics:
17. Borel subgroups and applications; 18. The geometry of algebraic groups; 19. Semisimple and reductive groups; 20. Algebraic groups of semisimple rank one; 21. Split reductive groups; 22. Representations of reductive groups; 23. The isogeny and existence theorems; 24. Construction of the semisimple groups; 25. Additional topics.

The main text is accompanied by three appendices providing some foundational material as freely used throughout the book.
Appendix A gives an overview of the basic definitions and facts from algebraic geometry, especially with a view toward the theory of schemes and their morphisms.
Appendix B discusses various existence theorems concerning quotients of algebraic groups modulo algebraic subgroups.
Finally, Appendix C reviews root systems and root data, that is, the combinatorial tools for the study of split reductive groups in Part D. These three appendices form the fifth Part E of the text. At the end of the book, there is a rich bibliography, followed by a detailed, carefully compiled index.
All together, this excellent text fills a long-standing gap in the textbook literature on algebraic groups. It presents the modern theory of group schemes in a very comprehensive, systematic, detailed and lucid manner, with numerous illustrating examples and exercises. It is fair to say that this reader-friendly textbook on algebraic groups is the long-desired modern successor to the old, venerable standard primers mentioned above.