The Atiyah-Patodi-Singer (APS) index theorem is a significant result in differential geometry and analysis, particularly for manifolds with boundary. This book provides a detailed exposition of the theorem, placing it within the framework of 'b'-geometry and related pseudodifferential operators on compact manifolds with boundary. The text begins with an introduction to the Atiyah-Singer index theorem, which serves as a foundation for understanding the APS theorem. The book then delves into the concepts of b-geometry, including exact b-metrics, b-tangent bundles, and associated differential operators. It explores spin structures, Clifford algebras, and the properties of Dirac operators on manifolds with boundary. The text also discusses the small b-calculus and the full calculus, covering topics such as the Mellin transform, analytic Fredholm theory, and the construction of parametrices. The book presents the relative index theorem, cohomology, and resolvent analysis, leading to the local index theorem and its applications. The APS theorem is shown to be a variant of the Atiyah-Singer theorem in the b-category, which involves manifolds with boundary and metrics with complete cylindrical ends. The proof of the APS theorem is given in a direct manner, emphasizing the analytic framework and the interplay between topological and geometric concepts. The text also includes a discussion of the eta invariant and its role in the index formula, as well as the spectral flow and its implications for the index theorem. The book concludes with a bibliography of relevant references and an index of notations. Overall, the text provides a comprehensive treatment of the APS index theorem, its generalizations, and its applications in geometry and analysis.The Atiyah-Patodi-Singer (APS) index theorem is a significant result in differential geometry and analysis, particularly for manifolds with boundary. This book provides a detailed exposition of the theorem, placing it within the framework of 'b'-geometry and related pseudodifferential operators on compact manifolds with boundary. The text begins with an introduction to the Atiyah-Singer index theorem, which serves as a foundation for understanding the APS theorem. The book then delves into the concepts of b-geometry, including exact b-metrics, b-tangent bundles, and associated differential operators. It explores spin structures, Clifford algebras, and the properties of Dirac operators on manifolds with boundary. The text also discusses the small b-calculus and the full calculus, covering topics such as the Mellin transform, analytic Fredholm theory, and the construction of parametrices. The book presents the relative index theorem, cohomology, and resolvent analysis, leading to the local index theorem and its applications. The APS theorem is shown to be a variant of the Atiyah-Singer theorem in the b-category, which involves manifolds with boundary and metrics with complete cylindrical ends. The proof of the APS theorem is given in a direct manner, emphasizing the analytic framework and the interplay between topological and geometric concepts. The text also includes a discussion of the eta invariant and its role in the index formula, as well as the spectral flow and its implications for the index theorem. The book concludes with a bibliography of relevant references and an index of notations. Overall, the text provides a comprehensive treatment of the APS index theorem, its generalizations, and its applications in geometry and analysis.