The ''law of the wall'' familiar from turbulent boundary layer theory, has been extended to turbulent flow along a sharp density interface. The effective gravity of such an interface is an additional independent variable, so that in nondimensional form the ''law of the interface'' becomes a one-parameter family of curves. A convenient label for each curve is its ''Keulegan parameter'' a nondimensional combination of inner variables, which also plays a role in the theory of interfacial stability. One important difference between the law of the wall and the law of the interface is that the velocity gradient at an interface is typically much less than the stress divided by the molecular viscosity. Even in the nonturblent core of an interface the effective viscosity is appreciably higher than the molecular viscosity. This finding is explained by the presence of viscous wavelets on the interface which contribute to momentum transfer through a combination of ''sheltering'' and the generation of an interfacial viscous boundary layer. A simple approximate theory based on these ideas reproduces the observed functional form of effective viscosity versus Keulegan parameter (a square root relationship for miscible fluids). At interfaces with free surface energy, for low relative velocities, capillarity plays an important role. Capillary-gravity waves of minimum celerity become key agents of momentum transfer, so that effective interface viscosity comes to depend on surface tension. The second major effect of variations of effective gravity is changing hydro-dynamic roughness. The extent of this influence is governed by the buoyancy flux, and it also occurs over solid walls, although this has so far been ignored in the literature. Evidence from a variety of sources now suggests that hydrodynamic ''roughness'' is determined by the energy balance of turbulence very close to the wall. A simple approximate theory based on this postulate reproduces qualitatively the observed interface roughness changes. Bulk relationships (drag, shear, and mass transfer coefficients) for interfaces may be determined by matching inner and outer laws in the conventional manner. The most practical independent variables for the presentation of such laws appear to be a gravity to viscous force ratio (rather than Reynolds number) and a Keulegan parameter (rather than Froude number), both based on bulk external variables.