The theme of this book is operator theory on C-algebras. The main novel tool employed is the concept of local multipliers. Originally devised by Elliott and Pedersen in the 1970's in order to study derivations and automorphisms, local multipliers of C-algebras were developed into a powerful device by the present authors in the 1990's. The book serves two purposes. The first part provides the reader - specialist and advanced graduate student alike - with a thorough introduction to the theory of local multipliers. Only a minimal knowledge of algebra and analysis is required, as the prerequisites in both non-commutative ring theory and basic C-algebra theory are presented in the first chapter. In the second part, local multipliers are used to obtain a wealth of information on various classes of operators on C-algebras, including (groups of) automorphisms, derivations, elementary operators, Lie isomorphisms and Lie derivations, as well as others. Many of the results appear in print for the first time. The authors have made an effort to avoid intricate technicalities thus some of the results are not pushed to their utmost generality. Several open problems are discussed, and hints for further developments are given.