One of the corner stones of loop quantum gravity (LQG) is the Ashtekar-Lewandowski representation, a Hilbert space representation of the basic kinematical variables of the theory. It is constructed without using any background geometric structure, and hence is diffeomorphism invariant. On the one hand, much of the subsequent developments in LQG depend on this representation. On the other hand it is well know from quantum field theory that generically there exist many inequivalent, and hence physically different, representations of a given algebra of basic variables. It is therefor an important question wether there exist other representations in the case of LQG. Surprisingly, one can show that the AL representation is the only background independent representation of the algebra of basic variables of LQG. In the talk I will review motivation, precise formulation, and idea of proof, of this uniqeness result.