An anomaly is the failure of classical symmetry to survive the process of quantization and regularization. The study of anomalies is the key to a deeper understanding of quantum field theory and has played an increasingly important role in the theory over the past 20 years. This text presentsall the different aspects of the study of anomalies in an accessible and self-contained way. Much emphasis is now being placed on the formulation of the theory using the mathematical ideas of differential geometry and topology. This approach is followed here, and the derivations and calculations aregiven explicitly as an aid to students. Topics discussed include the relevant ideas from differential geometry and topology and the application of these paths (path integrals, differential forms, homotopy operators, etc.) to the study of anomalies. Chapters are devoted to abelian and nonabeliananomalies, consistent and covariant anomalies, and gravitational anomalies. The comprehensive overview of the theory presented in this book will be useful to both students and researchers.