This book is a comprehensive introduction to the cohomology of groups, written by Kenneth S. Brown. It is intended for second-year graduate students who have some knowledge of algebra and topology. The book presents a balanced approach, combining algebraic and topological techniques. The first six chapters cover the basics of the subject, while the remaining four chapters are more specialized and reflect the author's research interests. The book assumes knowledge of basic algebra (groups, rings, modules, and tensor products) and algebraic topology (fundamental group, covering spaces, simplicial and CW-complexes, and homology). Some theorems, especially in later chapters, require more advanced topological knowledge, such as the Hurewicz theorem or Poincaré duality. Readers without the required background can take these theorems on faith. The book includes numerous exercises, some of which contain results referred to in the text. Exercises are marked with an asterisk if they are more difficult or require more background. The author thanks several individuals for their helpful comments on a preliminary version of the book. The book is organized into chapters covering various aspects of group cohomology, including homological algebra, homology and cohomology with coefficients, low-dimensional cohomology and group extensions, products, cohomology theory of finite groups, equivariant homology and spectral sequences, finiteness conditions, Euler characteristics, and Farrell cohomology theory. Each chapter includes detailed explanations, examples, and references to further reading. The book also includes a notation index and a comprehensive index for easy reference.This book is a comprehensive introduction to the cohomology of groups, written by Kenneth S. Brown. It is intended for second-year graduate students who have some knowledge of algebra and topology. The book presents a balanced approach, combining algebraic and topological techniques. The first six chapters cover the basics of the subject, while the remaining four chapters are more specialized and reflect the author's research interests. The book assumes knowledge of basic algebra (groups, rings, modules, and tensor products) and algebraic topology (fundamental group, covering spaces, simplicial and CW-complexes, and homology). Some theorems, especially in later chapters, require more advanced topological knowledge, such as the Hurewicz theorem or Poincaré duality. Readers without the required background can take these theorems on faith. The book includes numerous exercises, some of which contain results referred to in the text. Exercises are marked with an asterisk if they are more difficult or require more background. The author thanks several individuals for their helpful comments on a preliminary version of the book. The book is organized into chapters covering various aspects of group cohomology, including homological algebra, homology and cohomology with coefficients, low-dimensional cohomology and group extensions, products, cohomology theory of finite groups, equivariant homology and spectral sequences, finiteness conditions, Euler characteristics, and Farrell cohomology theory. Each chapter includes detailed explanations, examples, and references to further reading. The book also includes a notation index and a comprehensive index for easy reference.