This monograph provides a systematic treatment of the abstract theory of adjoint semigroups. After presenting the basic elementary results, the following topics are treated in detail: The sigma (X, X )—topology, —reflexivity, the Favard class, Hille—Yosida operators, interpolation and extrapolation, weak —continuous semigroups, the codimension of X in X , adjoint semigroups and the Radon—Nikodym property, tensor products of semigroups and duality, positive semigroups and multiplication semigroups. The major part of the material is reasonably self—contained and is accessible to anyone with basic knowledge of semi— group theory and Banach space theory. Most of the results are proved in detail. The book is addressed primarily to researchers working in semigroup theory, but in view of the "Banach space theory" flavour of many of the results, it will also be of interest to Banach space geometers and operator theorists.