This advanced monograph on Galois representation theory was written by one of the world's leading algebraists. Directed at mathematics students who have completed a graduate course in introductory algebraic topology, it offers a full treatment of the subject.The first four chapters cover characteristic classes of Galois representations whose values lie in mod 2 Galois abelian cohomology of groups, nonabelian cohomology of groups, characteristic classes of forms and algebras, and higher-dimensional characteristic classes of bilinear forms and Galois representations. Subsequent chapters explore stable homotopy and induced representations, explicit Brauer induction theory, and applications of explicit Brauer induction to Artin root numbers and local root numbers.