The success of Density Functional Theory (DFT) is partly due to that of simple approximations, such as the Local Density Approximation (LDA), which uses results of a model, the homogeneous electron gas, to simulate exchange-correlation effects in real materials. We turn this intuitive approximation into a general and in principle exact theory by introducing the concept of a connector: a prescription how to use results of a model system in order to simulate a given quantity in a real system. In this framework, the LDA can be understood as one particular approximation for a connector that is designed to link the exchange-correlation potentials in the real material to that of the model. Formulating the in principle exact connector equations allows us to go beyond the LDA in a systematic way. Moreover, connector theory is not bound to DFT, and it suggests approximations also for other functionals and other observables. We explain why this very general approach is indeed a convenient starting point for approximations. We illustrate our purposes with simple but pertinent examples.