In Banach space theory, the ``local theory'' refers to the collection of finite dimensional methods and ideas which are used to study infinite dimensional spaces (see e.g. [P4,TJ]). It is natural to try to develop an analogous theory in the recently developed category of operator spaces [BP,B1-2,BS,ER1-7,Ru]. The object of this paper is to start such a theory. We plan to present a more thorough discussion of the associated tensor norms in a future publication.