We consider the free boundary problem of two superimposed, immiscible, viscous, incompressible fluids. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously, and we study the Rayleigh-Taylor instability. The approach relies on the theory of maximal regularity. (Joint work with Jan Pruss.)