This chapter presents a comprehensive formulation of Wilhelm Ackermann’s class theory. After discussing his initial formulation, which lacked the Axiom of Regularity, the axiom is introduced, and from the resulting complete system—referred to as system A—the set theory needed for the purposes of the book is developed. In fact, the treatment is quite detailed and can serve as an introduction to set theory. Moreover, it provides the axiomatic foundations for the development of category theory, as it allows for the existence of classes that are not sets. The power of system A is demonstrated by deriving the Zermelo–Fraenkel system with the Axiom of Choice (ZFC). Starting with the usual basic notions, the chapter develops in detail the theory of ordinal numbers, the rudiments of cardinal arithmetic, and includes a proof of Zorn’s Lemma.