The Atiyah-Patodi-Singer (APS) Index Theorem is a significant result in differential geometry and topology, generalizing the classical Atiyah-Singer Index Theorem to manifolds with boundary. This book, authored by Richard B. Melrose, provides a detailed and expanded version of lecture notes from a course at MIT in 1991. The APS theorem is presented as a natural extension of the Atiyah-Singer theorem to the context of manifolds with boundary, specifically those with exact $b$-metrics, which are metrics that make the neighborhood of the boundary asymptotically cylindrical. The book begins with an introduction to the APS theorem and its proof, emphasizing its simplicity and generality. It then delves into the detailed proof, which is direct in two senses: the conceptual background is straightforward, and the terms in the final formula emerge naturally during the proof. The author argues that the APS theorem is essentially the Atiyah-Singer theorem in the $b$-category, suggesting that there may be other such theorems for manifolds with corners. The subsequent chapters flesh out the proof and explore various aspects of the theory, including spin structures, small $b$-calculus, full calculus, relative index, cohomology, resolvent, heat calculus, local index theorem, and applications. The book also discusses extensions and generalizations of the APS theorem, such as those by Bismut, Cheeger, Moscovici, and Stern. The text is intended for readers with a background in pseudodifferential operators and Hodge theory, and it aims to provide a comprehensive understanding of the analytic framework in which the APS theorem is situated. It includes a bibliography and indices for notations, authors, and topics.The Atiyah-Patodi-Singer (APS) Index Theorem is a significant result in differential geometry and topology, generalizing the classical Atiyah-Singer Index Theorem to manifolds with boundary. This book, authored by Richard B. Melrose, provides a detailed and expanded version of lecture notes from a course at MIT in 1991. The APS theorem is presented as a natural extension of the Atiyah-Singer theorem to the context of manifolds with boundary, specifically those with exact $b$-metrics, which are metrics that make the neighborhood of the boundary asymptotically cylindrical. The book begins with an introduction to the APS theorem and its proof, emphasizing its simplicity and generality. It then delves into the detailed proof, which is direct in two senses: the conceptual background is straightforward, and the terms in the final formula emerge naturally during the proof. The author argues that the APS theorem is essentially the Atiyah-Singer theorem in the $b$-category, suggesting that there may be other such theorems for manifolds with corners. The subsequent chapters flesh out the proof and explore various aspects of the theory, including spin structures, small $b$-calculus, full calculus, relative index, cohomology, resolvent, heat calculus, local index theorem, and applications. The book also discusses extensions and generalizations of the APS theorem, such as those by Bismut, Cheeger, Moscovici, and Stern. The text is intended for readers with a background in pseudodifferential operators and Hodge theory, and it aims to provide a comprehensive understanding of the analytic framework in which the APS theorem is situated. It includes a bibliography and indices for notations, authors, and topics.