This advanced algebra book deals with derivations, generalized derivations, centralizers and theta-centralizers. It studies prime and semiprime rings. It has seven chapters. Chapter one gives the preliminaries on derivations, generalized derivations, centralizers and theta centralizers. Chapter two presents additivity results for multiplicative generalized derivations and multiplicative left centralizers. Chapter three extends Posner''s first Theorem with generalized derivations on Lie ideals in prime rings. Chapter four presents a proof which shows that in a semiprime ring, under some conditions, any Jordan generalized derivation must be a generalized derivation. We include identities which force additive mappings to be generalized derivations and Jordan *-generalized derivation, where * is the involution mapping. Chapter five characterizes rings with a Jordan centralizer . Chapter six displays the identities which force additive mappings to be theta-centralizers. Chapter seven deals with free action mappings and the dependent elements related to those mappings. It also gives a generalization of the definition of dependent elements and free actions.