Surface meltwater transport on the Greenland Ice Sheet has important implications for mass balance. Melt runoff is the principle mechanism for loss in the surface mass balance, and any meltwater that reaches the englacial and subglacial hydraulic systems can contribute to sliding and dynamic losses from the ice sheet. In the ablation zone of an ice sheet, all melt is runoff, but in the accumulation zone, some melt is retained each year by percolation into the underlying cold snow and subsequent refreezing of that melt. The runoff limit is probably located some distance above the equilibrium line around the soaked snow line, but it has never been directly measured. The amount of downslope flow and the lateral distance it can travel are required for runoff calculations. Two models were developed to address the issues of downslope melt transport in the accumulation zone of Greenland. A one-dimensional model simulates the effects multi-year cycles of melt, infiltration and refreeze have on snowpack physical properties and downslope flow of meltwater. It was found that increasing accumulation during the melt season led to a greater fraction of surface melt traveling downslope, while higher winter accumulation leads to more vertical infiltration and less downslope flow. The temperature conditions in the snowpack control the lateral distance this melt can travel. This model also indicates that the fraction of melt that is runoff increases with decreasing elevation on the ice sheet, and there is likely not one elevation that acts as a threshold for runoff. In the area of the West Greenland Ice Sheet in the current climate, the maximum distance melt can travel downslope is about 12km. The two-dimensional model simulates the details of unsaturated meltwater flow and energy balance very low in the accumulation zone. Layering in the two-dimensional model increases downslope flow, and meltwater is able to travel over snow layers that are permeable. Saturated conditions can occur over unsaturated layers when the surface melt flux is large enough, which means the entire pore space of the snow column does not need to become saturated before downslope flow occurs. Significant lateral transport of meltwater does not occur until the snow layers are saturated, which requires that very little refreezing takes place in the snowpack. Thus the snow must become temperate or nearly so before surface melt can travel downslope enough to become runoff. This occurs below the saturation line in the soaked snow facies of the Greenland Ice Sheet.