This paper introduces a methodology for the symbolic control of nonlinear systems based on an approximate notion of simulation relation. This notion generalizes existing exact notions of simulation and is completely characterized in terms of known stabilizability concepts. Equipped with this notion we show how, under certain stabilizability assumptions, we can construct finite or symbolic models for nonlinear control systems. Synthesizing controllers for the original control system can then be done by using supervisory control techniques on the finite models and by refining the resulting finite controllers to hybrid controllers enforcing the specification on the original continuous control system. The proposed design methodology can be seen as a correct-by-design way of obtaining both the feedback control laws as well as the control software responsible for deciding which law is executed and when.