This comprehensive textbook was written for students and lecturers in mathematics and computer science, and due to its detailed presentation of Gödel's incompleteness theorems, it is also suitable for students specializing in philosophical logic. For this new edition, the text has been thoroughly revised both factually and stylistically. It includes improved proofs and exercises with hints for solutions, as well as a historically oriented introduction. The presentation is now better tailored to Bachelor/Master programs in mathematics and computer science. The book can be used independently of lectures, as well as for self-study.