Ordinal numbers are transfinite generalisations of natural numbers, and are usually defined and studied concretely as special types of sets. In this thesis we explore an abstract approach to developing the theory of ordinal numbers, where we present various axiomatisations of an ordinal number system and prove their equivalence. Since ordinal numbers do not form a set, in order to develop such a theory one needs to extend the usual framework of Zermelo-Fraenkel set theory. Among several such possible extensions, we pick the one that is based on the notion of a Grothendieck universe. While some of the results obtained in this thesis are merely adaptations of known results to this context, some others are new even to classical set theory. Among these is a definition and a universal property of the ordinal number system that mimics the classical Dedekind-Peano approach to the natural number system.