In many physical systems it is believed that one can describe parts of the system by classical and parts of the system by quantum mechanics. In this lecture I explain how to rigorously derive such effective "mixed quantum-classical" descriptions from the underlying quantum mechanics for the whole system. The important mechanism is adiabatic decoupling and the method adiabatic perturbation theory. As the paradigmatic example I will discuss the time-dependent Born-Oppenheimer approximation.