'"Models of Peano arithmetic" is a book that should have been written many years ago ... the subject has never had a standard introductory text ... For many technical reasons potential authors have found the task of writing such a text rather difficult, leaving this interesting area of research without the presentation it deserves. Kaye's book fills this gap in literature remarkably well. The presentation will certainly satisfy all who ever wondered what such a formula looks like.'R. Kossak, Zentralblatt für Mathematik und ihre Grenzgebiete Mathematics Abstracts'It is carefully written and therefore recommended to anyone who wants to learn about this theorem.'A. Di Bucchianico, MWG, No. 4, April 1993'Kaye's book has many obvious virtues. It covers a great number of topics which are of the utmost importance for metamathematics of PA. Thus, the book is an advanced monograph which offers a unified treatment of various aspects of the first-order PA ... the author has succeeded in producing a text which can also serve as a graduate textbook of metatheory of PA; the book is clearly written and contains several well-motivated exercises.'Jan Wole'nski, Jagiellonian University, Poland'Its well-written presentaiton is aimed at a general (mathematical)audience.'P. Schmitt, Monatshefte fur Mathematik