I first seriously contemplated writing a book on degree theory in 1976 while I was visiting the University of Illinois at Chicago Circle. There was, at that time, some interest in an Q-series book about degree theory, and through the encouragement of Bob Soare, I decided to make a proposal to write such a book. Degree theory had, at that time, matured to the point where the local structure results which had been the mainstay of the earlier papers in the area were finding a steadily increasing number of applications to global degree theory. Michael Yates was the first to realize that the time had come for a systematic study of the interaction between local and global degree theory, and his papers had a considerable influence on the content of this book.

During the time that the book was being written and rewritten, there was an explosion in the number of global theorems about the degrees which were proved as applications of local theorems. The global results, in turn, pointed the way to new local theorems which were needed in order to make further progress. I have tried to update the book continuously, in order to be able to present some of the more recent results. It is my hope to introduce the reader to some of the fascinating combinatorial methods of Recursion Theory while simultaneously showing how to use these methods to prove some beautiful global theorems about the degrees.

This book has gone through several drafts. An earlier version was used for a one semester course at the University of Connecticut during the Fall Semester of 1979, at which time a special year in Logic was taking place. Many helpful comments were received from visitors to UConn and UConn faculty at that time. Klaus Ambos, David Miller and James Schmerl are to be thanked for their helpful comments. Steven Brackin and Peter Fejer carefully read sizable portions of that version and supplied me with many corrections and helpful suggestions on presentation. Richard Shore, Stephen Simpson and Robert Soare gave helpful advice about content and presentation of material. Other people whose comments, corrections and suggestions were of great help are Richard Epstein, Harold Hodes, Carl Jockusch, Jr. Azriel Levy and George Odifreddi.