Flows of sediment-water mixture (suspensions) are common superficial processes in nature. To investigate the possibility of predicting the behavior of such flows, we study the free surface flow of a clay-water suspension down a slope, which may represent a simple realistic model for the flow of some concentrated sediment-water mixture in a natural environment. The hydrodynamic lubrication equations governing the flow are derived from the full Navier-Stokes equations, including an appropriate tensorial expression for the constitutive equation of the suspension (a Herschel-Bulkley fluid with a hysteresis in the yield strength). Results have been obtained for the free surface and length of an unsteady, laminar, isothermal flow of the suspension. The theory is in good agreement with laboratory experiments with fine clay suspensions. ¿ 2000 American Geophysical Union