Designed for advanced undergraduate or beginning graduate students, this book provides a complete introduction to computability theory. This second edition includes new material on hyperarithmetical and Borel sets as well as more material on computability of structures, Pi-0-1 classes, and computability in science. It features an expanded treatment of complexity of computations and updated future directions in computability. In addition, the section on randomness is now a separate chapter. The author also discusses advanced topics in greater depth, including Post's problem, forcing and category, applications of determinacy, and the computability of theories.