The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. At the same time, it is often viewed as a foundation of mathematics so that in the most prevalent, current mathematical practice "to make a notion precise" simply means "to define it in set theory." This book tries to do justice to both aspects: it gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation. The book is geared to advanced undergraduate or beginning graduate mathematics students and mathematically minded graduate students in computer science and philosophy.