Observations of the flow of dense fluid into uniformly density-stratified environments down plane slopes with small inclination to the horizontal ([les ] 20°) are described, and a quantitative model for such flows is presented. In these experiments the dense fluid is released at the top of the slope for a finite period of time. The resulting downslope gravity current, or downflow, has uniform thickness with a distinct upper boundary, until it approaches its level of neutral density where the fluid leaves the proximity of the slope. Turbulent transfers of mass and momentum occur across the upper boundary, causing a continuous loss of fluid from the downflow in most cases, and associated loss of momentum. The flow may be characterized by the local values of the Richardson number Ri, the Reynolds number Re (generally large), and of M = QN3/g′2, where Q is the (two-dimensional) volume flux, N the buoyancy frequency and g′ the (negative) buoyancy of the dense fluid. The model for the downflow describes the turbulent transfers in terms of entrainment, detrainment and drag coefficients, Ee, Ed and k respectively, and the observations enable the determination of these coefficients in terms of the local values of M and Ri. The model may be regarded as an extension of that Ellison & Turner (1959) to stratified environments, describing the consequent substantial changes in mixing and distribution of the inflow. It permits the modelling of the bulk properties of these flows in geophysical situations, including shallow and deep flows in the ocean.