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. (…)