The highland valley networks are perhaps the most compelling evidence for widespread fluvial activity on Mars >3.5 Ga. However, determining the hydrology of these features has been difficult owing to poor image resolution and the lack of available topographic data. New orbital imaging reveals 21 late-stage channels within valley networks, which we use to estimate formative discharges and to evaluate water supply mechanisms. We find that channel width and associated formative discharge are comparable to terrestrial valley networks of similar area and relief. For 15 narrow channels in basin-filling networks, likely episodic runoff production rates up to centimeters per day and first-order formative discharges of ∼300–3000 m3/s are similar to terrestrial floods supplied by precipitation. Geothermal melting of ground ice would produce discharges ∼100 times smaller per unit area and would require pulsed outbursts to form the channels. In four large valleys with few tributaries, wider channels may represent large subsurface outflows or paleolake overflows, as these four channels originate at breached basin divides and/or near source regions for the catastrophic outflow channels.