When I was a beginning mathematics student a friend gave me a set of lecture notes for a course on infinitary logic given by Ronald Jensen. On the first page was the definition of a partial isomorphism: a set of partial mappings between two structures with the back-and-forth property. I became immediately interested and now – 37 years later – I have written a book on this very concept.

This book can be used as a text for a course in model theory with a game- and set-theoretic bent.

I am indebted to the students who have given numerous comments and corrections during the courses I have given on the material of this book both in Amsterdam and in Helsinki. I am also indebted to members of the Helsinki Logic Group, especially Tapani Hyttinen and Juha Oikkonen, for discussions, criticisms, and new ideas over the years on Ehrenfeucht–Fraïssé Games in uncountable structures. I am grateful to Fan Yang for reading and commenting on parts of the manuscript.

I am extremely grateful to my wife Juliette Kennedy for encouraging me to finish this book, for reading and commenting on the manuscript pointing out necessary corrections, and for supporting me in every possible way during the writing process.

The preparation of this book has been supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I am grateful to the Institute for Advanced Study, the Mittag-Leffler Institute, and the Philosophy Department of Princeton University for providing hospitality during the preparation of this book.