This is a very interesting book on an important topic both for physics and for mathematics. (…) It starts at the beginning, but is not really for beginners; the physics background develops rapidly, through seven short chapters, and the final eight chapters provide a lightning review of the mathematical topics encountered (…) Part II is the main part of the text, containing a selection of fascinating topics, beautifully presented, to many of which the author has been a significant contributor. The chapters on functional integration, on elliptic operators, their determinants and related index theorems, on calculating instanton contributions and on anomalies are particularly attractive. (…) Bulletin London Mathematical Society