This monograph is concerned with the interplay between the theory of operator semigroups and spectral theory. The basics on operator semigroups are concisely covered in this self-contained text. Part I deals with the Hille--Yosida and Lumer--Phillips characterizations of semigroup generators, the Trotter--Kato approximation theorem, Kato\'s unified treatment of the exponential formula and the Trotter product formula, the Hille--Phillips perturbation theorem, and Stone\'s representation of unitary semigroups. Part II explores generalizations of spectral theory\'s connection to operator semigroups. XIV, 266 p.