The spreading of a two-dimensional, viscous gravity current propagating over and draining into a deep porous substrate is considered both theoretically and experimentally. We first determine analytically the rate of drainage of a one-dimensional layer of fluid into a porous bed and find that the theoretical predictions for the downward rate of migration of the fluid front are in excellent agreement with our laboratory experiments. The experiments suggest a rapid and simple technique for the determination of the permeability of a porous medium. We then combine the relationships for the drainage of liquid from the current through the underlying medium with a formalism for its forward motion driven by the pressure gradient arising from the slope of its free surface. For the situation in which the volume of fluid V fed to the current increases at a rate proportional to t3, where t is the time since its initiation, the shape of the current takes a self-similar form for all time and its length is proportional to t2. When the volume increases less rapidly, in particular for a constant volume, the front of the gravity current comes to rest in finite time as the effects of fluid drainage into the underlying porous medium become dominant. In this case, the runout length is independent of the coefficient of viscosity of the current, which sets the time scale of the motion. We present numerical solutions of the governing partial differential equations for the constant-volume case and find good agreement with our experimental data obtained from the flow of glycerine over a deep layer of spherical beads in air.