If the large Late Hesperian outflow channels were eroded by extensive floods, as appears likely, then large bodies of water must have once occupied the northern plains during that period. Previous estimates of the sizes of bodies of water in the northern lowlands range up to 3 ?? 108 km3. Several contacts have been previously mapped around the edges of the northern plains and interpreted to be shorelines remaining from these former standing bodies of water. We examine the elevations and geologic relations along these contacts in detail and find little support for their interpretation as shorelines. Some contacts are clearly of volcanic origin, and all have significant variations in elevation. Better support for the former presence of water over large parts of the northern plains is provided by the Vastitas Borealis Formation (VBF). Most of the post-Noachian fill within the northern basin is ridged plains of Lower Hesperian age, interpreted to be volcanic in origin. Overlying the ridged plains is the VBF, a thin veneer of material of Upper Hesperian age. The VBF may have been deposited from large floods. Support for this interpretation is the similarity in age between the outflow channels and the VBF, the presence of the VBF at the lower ends of the outflow channels, and identification of numerous features in the outcrop areas of the VBF that are suggestive of basal melting of an ice sheet. To cover all the area over which the VBF is exposed would require ???2.3 ?? 107 km3 of water. Spread over the entire surface of Mars, this volume is equal to a global layer (global equivalent layer, or GEL) ???156 m deep. We find no support for the larger estimates of ocean volumes that range up to 3 ?? 108 km3 and which imply comparable amounts of water per unit area as are currently present on the surface of the Earth. Under present climatic conditions on Mars an ocean would freeze in a geologically short time period (???104 years), then would sublimate away at rates strongly dependent on the presence or absence of debris on the ice surface. The present VBF is interpreted as a sublimation residue from the ponded outflow channel effluents. The fate of a volume of water thought to have been emplaced by the outflow channels (???2.3 ?? 107 km3) is largely accounted for by the presence of other existing reservoirs on the planet. An approximately 20-30 m GEL of water is estimated to be in the present polar caps, and a 50 m GEL may have escaped to space since the Hesperian, leaving ???80 m GEL unaccounted for. This amount may be partly trapped in other volatile-rich deposits on the surface, and a significant amount could have reentered the groundwater system by south polar basal melting and been progressively cold-trapped at the base of a growing cryosphere. On the basis of our assessment of the Hesperian-aged deposits, we predict that testing of the Clifford and Parker  hypothesis that a Noachian-aged ocean covered up to one third of the surface of Mars will be made very difficult by the enhanced degradation rates in the Noachian and subsequent geological events in the northern lowlands.