This book, "Loop Spaces, Characteristic Classes and Geometric Quantization" by Jean-Luc Brylinski, is a comprehensive treatment of characteristic classes and their applications in various areas of mathematics and physics. The author develops a theory of characteristic classes for gerbes, which are fiber bundles over a manifold with groupoid fibers. This theory generalizes the classical theory of characteristic classes for vector bundles and principal bundles, providing a geometric framework for understanding higher-dimensional cohomology classes. The book is structured into seven chapters, each focusing on different aspects of the theory. Chapter 1 introduces sheaves, complexes of sheaves, and their cohomology. Chapter 2 discusses line bundles, their classification, and the relationship between line bundles and degree 2 cohomology. Chapter 3 explores the geometry of the space of singular knots, showing that it has a structure similar to a Kähler manifold. Chapter 4 presents the first level of the theory of degree 3 cohomology, based on infinite-dimensional algebra bundles. Chapter 5 delves into the second level of the theory using sheaves of groupoids, defining the concept of 3-curvature and identifying the group of equivalence classes of Dixmier-Douady sheaves of groupoids with Deligne cohomology. Chapter 6 constructs line bundles on the free loop space and the space of knots, using the 3-curvature of sheaves of groupoids. Chapter 7 discusses the Dirac magnetic monopole and the quantization condition for closed 3-forms on the 3-sphere. The book aims to provide a geometric interpretation of degree 3 cohomology and its applications, such as in geometric quantization, knot theory, and gauge theory. It is intended for mathematicians and physicists working in topology, geometry, and mathematical physics, requiring a background in point-set topology, differential geometry, and graduate algebra.This book, "Loop Spaces, Characteristic Classes and Geometric Quantization" by Jean-Luc Brylinski, is a comprehensive treatment of characteristic classes and their applications in various areas of mathematics and physics. The author develops a theory of characteristic classes for gerbes, which are fiber bundles over a manifold with groupoid fibers. This theory generalizes the classical theory of characteristic classes for vector bundles and principal bundles, providing a geometric framework for understanding higher-dimensional cohomology classes. The book is structured into seven chapters, each focusing on different aspects of the theory. Chapter 1 introduces sheaves, complexes of sheaves, and their cohomology. Chapter 2 discusses line bundles, their classification, and the relationship between line bundles and degree 2 cohomology. Chapter 3 explores the geometry of the space of singular knots, showing that it has a structure similar to a Kähler manifold. Chapter 4 presents the first level of the theory of degree 3 cohomology, based on infinite-dimensional algebra bundles. Chapter 5 delves into the second level of the theory using sheaves of groupoids, defining the concept of 3-curvature and identifying the group of equivalence classes of Dixmier-Douady sheaves of groupoids with Deligne cohomology. Chapter 6 constructs line bundles on the free loop space and the space of knots, using the 3-curvature of sheaves of groupoids. Chapter 7 discusses the Dirac magnetic monopole and the quantization condition for closed 3-forms on the 3-sphere. The book aims to provide a geometric interpretation of degree 3 cohomology and its applications, such as in geometric quantization, knot theory, and gauge theory. It is intended for mathematicians and physicists working in topology, geometry, and mathematical physics, requiring a background in point-set topology, differential geometry, and graduate algebra.