This book is an advanced, comprehensive text covering branches of the problems of models of Peano arithmetic that are not treated in Kayes’s book, “Models of Peano arithmetic”, a ‘bible’ of the study of models of arithmetic. It is assumed that the reader is familiar with Kaye’s book. Roughly speaking, the two basic themes of the book are the substructure lattices of elementary submodels of a given model of Peano arithmetic and recursive and arithmetic saturation. The first theme is mostly developed in the first four chapters (extensions in chapter 2, minimal and other types in chapter 3 and interstructure lattices in chapter 4). The second main topic can be found in chapters 5–11. Automorphism groups of recursively saturated models are studied in chapters 8 and 9, indiscernible generators are treated in chapter 5, the ω1-like models in chapter 10 and a classification of reducts in chapter 11. Chapter 6 deals with generics and forcing. ‘Twenty questions’ is the name of the last chapter. Exercises of various difficulties are included as an integral part of each chapter. The book is a welcome, comprehensive and useful publication.