This book introduces the most important and useful algebraic and topological techniques for studying and calculating the cohomology of finite groups. Numerous applications are described, to problems in homotopy theory, representation theory, number theory and group actions. Significant calculations are included with complete details and explanations.

I. Group Extensions, Simple Algebras and Cohomology.- II. Classifying Spaces and Group Cohomology.- III. Modular Invariant Theory.- IV. Spectral Sequences and Detection Theorems.- V. G-Complexes and Equivariant Cohomology.- VI. The Cohomology of Symmetric Groups.- VII. Finite Groups of Lie Type.- VIII. Cohomology of Sporadic Simple Groups.- IX. The Plus Construction and Applications.- X. The Schur Subgroup of the Brauer Group.- References.