Aqueous foams are complex fluids composed of gas bubbles tightly packed in a surfactant solution. Even though they generally consist only of Newtonian fluids, foam flow obeys nonlinear laws. This can result from nonaffine deformations of the disordered bubble packing as well as from a coupling between the surface flow in the surfactant monolayers and the bulk liquid flow in the films, channels, and nodes. A similar coupling governs the permeation of liquid through the bubble packing that is observed when foams drain due to gravity. We review the experimental state of the art as well as recent models that describe the interplay of the processes at multiple length scales involved in foam drainage and rheology.