Publisher Summary This chapter discusses a theory of generalized identities (GIs). Not only is this theory very pretty and natural, arising merely by extending the set of coefficients, but it provides a very useful way to encode specific information about rings and their elements. Several important polynomial identities (PI)-theorems turn out to be consequences of more general, easy-to-prove facts in the GI-theory. The heart of GI-theory lies in the structure of primitive rings. The chapter discusses most of the major structure theorems of GI-theory, stemming from a technical result describing the “evaluations” of a generalized monomial on a primitive ring.