Eugène Charles Catalan made his famous conjecture that 8 and 9 are the only two consecutive perfect powers of natural numbers in 1844 in a letter to the editor of Crelles mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it. Catalans Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The author dissects both Mihailescus proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaines theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further. Beautifully clear and concise, this book will appeal not only to specialists in number theory but to anyone interested in seeing the application of the ideas of algebraic number theory to a famous mathematical problem.