Geological evidence for extensive non-polar ice deposits of Amazonian age indicates that the current cold and dry climate of Mars has persisted for several billion years. The geological record and climate history of the Noachian, the earliest period of Mars history, is less certain, but abundant evidence for fluvial channels (valley networks) and lacustrine environments (open-basin lakes) has been interpreted to represent warm and wet conditions, including rainfall and runoff. Alternatively, recent atmospheric modeling results predict a “cold and icy” Late Noachian Mars in which moderate atmospheric pressure accompanied by a full water cycle produce an atmosphere where temperature declines with elevation following an adiabatic lapse rate, in contrast to the current situation on Mars, where temperature is almost completely determined by latitude. These results are formulated in the Late Noachian Icy Highlands (LNIH) model, in which these cold and icy conditions lead to the preferential deposition of snow and ice at high elevations, such as the southern uplands. What is the fate of this snow and ice and the nature of glaciation in such an environment? What are the prospects of melting of these deposits contributing to the observed fluvial and lacustrine deposits?To address these questions, we report on a glacial flow-modeling analysis using a Mars-adapted ice sheet model with LNIH climate conditions. The total surface/near-surface water inventory is poorly known for the Late Noachian, so we explore the LNIH model in a “supply-limited” scenario for a range of available water abundances and a range of Late Noachian geothermal fluxes. Our results predict that the Late Noachian icy highlands (above an equilibrium line altitude of approximately +1km) were characterized by extensive ice sheets of the order of hundreds of meters thick. Due to extremely cold conditions, the ice-flow velocities in general were very low, less than a fewmm/yr, and the regional ice-flow pattern was disorganized and followed topography, with no radial flow pattern typical of an equilibrium ice sheet. Virtually the entire ice sheet is predicted to be cold-based, and thus the range of wet-based features typically associated with temperate glaciers (e.g., drumlins, eskers, etc.) is not predicted to occur. Wet-based conditions are predicted only locally in the thickest ice (on the floors of the deepest craters), where limited subglacial lakes may have formed.These LNIH regional ice-sheets provide a huge reservoir of potential meltwater as a source for forming the observed fluvial and lacustrine features and deposits. Top-down melting scenarios applied to our LNIH ice sheet model predict that periods of punctuated warming could lead to elevated temperatures sufficient to melt enough snow and ice to readily account for the observed fluvial and lacustrine features and deposits. Our model indicates that such melting should take place preferentially at the margins of the ice sheets, a prediction that can be tested with further analyses.