The book under review is a monograph devoted to (nonstandard) models of first-order arithmetic. Its main aim is the description of the structure of such models. The book consists of 11 chapters. After having introduced in Chapter 1 the basic notions, the following subjects are considered: extensions of models, minimal and other types, substructure lattices, the control of types, generics and forcing, cuts, automorphisms and automorphism groups of recursively saturated models, \(\omega_{1}\)-like models, order types. The book is closed by the chapter “Twenty questions”, in which the authors give 20 open questions. A note added in proof says that two of them have been solved after the list have been circulated among the authors’ colleagues. Each chapter is finished by Remarks and References where the authors attempt to provide accurate credits and references to results presented in the given chapter. An integral part of the book are exercises. They have been divided into four categories: traditional exercices for practice, exercises containing lemmas, propositions and theorems taken from the literature, and the most difficult exercises that the authors “could not do” (p. ix). The book contains also an extensive list of references (for each item a number indicating the page number where it is cited is given) as well as an index of names and an index of terms. Most of the results presented in the book have been previously published, but many proofs are new or simpler or much shorter than the original ones. Some results appear for the first time. The advantage of the book is that it provides the whole range of various techniques presented in a unified way. The book is well written and it has been published in a very careful way. It is an excellent book and will certainly become one of the books that everyone working in the field of models of arithmetic should have and read. It supplements the book by R. Kaye [Models of Peano arithmetic. Oxford: Clarendon Press (1991; Zbl 0744.03037)] and the book by P. Hájek and P. Pudlák [Metamathematics of first-order arithmetic. Berlin: Springer-Verlag (1993; Zbl 0781.03047)].