Large cardinal hypotheses play a central role in modern set theory. One important way to understand such hypotheses is to construct concrete, minimal universes, or core models , satisfying them. Since Goedel’s pioneering work on the universe of constructible sets, several larger core models satisfying stronger hypotheses have been constructed, and these have proved quite useful. Here the author extends this theory so that it can produce core models satisfying There is a Woodin cardinal , a large cardinal hypothesis which is the focus of much current research. The book is intended for advanced graduate students and reseachers in set theory.