This book is an improved and much expanded version of a long unavailable University of Warwick preprint (An Equivariant Kunneth Formula in K-theory, Warwick 1967; referred to as [0] in this introduction). The aims are to prove the existence of the equivariant Kunneth formula; to situate the proof in a general theory of Eilenberg-Moore sequences; and to use the formula to obtain interesting applications.