This book reviews the present state of knowledge of the anomalous magnetic moment a=(g-2)/2 of the muon. The muon anomalous magnetic moment amy is one of the most precisely measured quantities in elementary particle physics and provides one of the most stringent tests of relativistic quantum field theory as a fundamental theoretical framework. It allows for an extremely precise check of the standard model of elementary particles and of its limitations. Recent experiments at the Brookhaven National Laboratory now reach the unbelievable precision of 0.5 parts per million, improving the accuracy of previous g-2 experiments at CERN by a factor of 14. A major part of the book is devoted to the theory of the anomalous magnetic moment and to estimates…