The outstanding properties of this book are lively, clear exposition, and discussion of a wide range of applications of path integration. The applications include the standard ones of the perturbation expansion in nonrelativistic quantum mechanics, the WKB approximation, and the polaron (one of the early successes of the path integral method). Unusual applications include the discussion of spin, path integrals for multiply connected spaces, black holes, and critical droplets. Where needed, brief discussions of mathematical topics, such as the Morse index theorem and homotopy theory, which might not be familiar to the audience of physicists and applied mathematicians to which the book is addressed, are inserted. Quantum field theory is treated only in a cursory way.