Path integrals provide a powerful method for describing quantum phenomena. This book introduces the quantum mechanics of particles moving in curved space by employing path integrals and then using them to compute anomalies in quantum field theories. These anomalies provide enormous constraints in the search for physical theories such as those of elementary particles, quantum gravity and string theories. An advanced text for researchers and graduate students of quantum field theory and string theory, the first part is also a stand-alone introduction to path integrals in quantum mechanics.