Inductive limits are a central construction in operator algebras because they allow one to use well-understood building blocks to naturally construct more complicated objects whose properties remain tractable. In the classical setting, few nuclear C*-algebras arise as inductive limits of finite-dimensional C*-algebras with *-homomorphism connecting maps. However, by generalizing our notion of an inductive system to one with completely positive contractive connecting maps, we can in fact realize any nuclear C*-algebra as the limit of a system of finite dimensional C*-algebras. Moreover, we can give necessary and sufficient conditions for the limit of such a system to be a (nuclear) C*-algebra. These systems arise naturally from completely positive approximations of nuclear C*-algebras. This is based in part on joint work with Wilhelm Winter.