The study of polynomial completeness of algebraic systems has only recently matured, and until now, lacked a unified treatment. Polynomial Completeness in Algebraic Systems examines the entire field with one coherent approach. The authors focus on the theory of affine complete varieties but also give the primary known results on affine completeness in special varieties. The book includes an extensive introductory chapter that provides the necessary background and makes the results accessible to graduate students as well as researchers. Numerous exercises illustrate the theory, and examples-and counterexamples-clarify the boundaries of the subject.