This book contains the results of the research of the authors in the foundations of the theory of definable actions of Polish groups and the associated orbit equivalence relations. It is addressed to graduate students and researchers in this area as well as to mathematicians in other fields, whose work comes in contact with this subject.

The reader will also find some new theorems which are attributed to someone other than the authors. Many people have helped in one way or another to improve this book: by allowing us to include these new theorems; by simplifying some of our proofs; by bringing to our attention some work we were unaware of; or by reading earlier versions of this book – or parts of it – and suggesting improvements in the exposition. We thank all of them: G. Cherlin, W. Comfort, R. Dougherty, G. Hjorth, A. Louveau, M. Megrelishvili, M. G. Nadkarni, R. Sami, S. Solecki, S. Thomas, and an anonymous referee.

The authors would also like to acknowledge the financial support of the National Science Foundation, through grants DMS–9206922 and DMS– 9317509, and the help of the Mathematics Department at Caltech in the preparation of this book. We particularly thank J. Cassidy for typing the manuscript.