This is the first comprehensive monograph on the mathematical theory of the solitaire game The Tower of Hanoi. It comprises a survey of the historical development from its invention in 1883 by the French number theorist idouard Lucas to recent research in mathematics and applications in computer science and psychology. For instance, the so-called Frame-Stewart conjecture is an open problem since almost 70 years and shows the timeliness of the topic. The bookcontains a thorough presentation of the essential mathematical results with complete proofs. The main objects of research today are the so-called Hanoi graphs and the related SierpiÅski graphs. The latter have been introduced by Sandi KlavÅ¾ar und UroÅ¡ Milutinovic.Algorithms (with proofs of correctness) will be an essential part of the book, because of the great popularity of the topic in computer science. In view of the most important practical application of The Tower of Hanoi and related puzzles, namely in cognitive (neuro-)psychology, we will also address variants employed in test tools and analyze their mathematical structure.