Independently contacted coupled quantum wells separated by barriers which allow significant interlayer interactions but no tunneling have been fabricated. When current is passed through one layer, the interlayer interactions drag carriers in the second layer, resulting in a voltage response (for open circuits). The magnitude of the response gives a quantitative measure of the effective interlayer interactions and response functions of the system, and hence this is an excellent laboratory for the study of many-body phenomena in two-dimensional electron gases. We review the Boltzmann and Kubo formalisms for the theory of drag effects in coupled quantum wells and discuss three specific cases where many-body effects significantly affect the drag: (1) acoustic phonon-mediated drag, (2) large enhancements due to coupled plasmon modes, and (3) interplay of screening and Landau levels in large magnetic fields.