It is generally well known that there is an innate notion of things like isomorphisms or epimorphisms. This allows us to talk about isomorphisms or epimorphisms of various objects: groups, rings, algebras,etc. In other words, “isomorphism” is really a categorical notion.However, it is not so well known that finiteness itself is alsocategorical. In this talk, we will discuss how finiteness applies tovarious categories. This will allow usto see finite sets, finite dimensional vector spaces, finitely generated algebras and compact sets as manifestations of the same basic idea.