For a photocatalytic reaction to be thermodynamically allowed, a semiconductor's band edges need to be placed appropriately relative to the reaction redox potentials. We apply a recently developed scheme for calculating band edges with density functional theory (DFT)-based methods to Cu2O, evaluating its available thermodynamic overpotential for redox reactions such as water splitting and conversion of CO2 to methanol. Because these calculations are surface dependent, we first study the low-index surfaces of Cu2O using periodic DFT+U theory to characterize and identify the most stable surface, which will be the most catalytically relevant. We employ various techniques to calculate the surface energy, including the method of "ab initio atomistic thermodynamics". The Cu2O(111) surface with (1 x 1) periodicity and surface copper vacancies is identified as the most stable at all oxygen partial pressures, although the ideal stoichiometric Cu2O(111) surface is relatively close in energy under oxygen-poor conditions. These surfaces are then used to calculate the band edges. Comparison of the band edges to redox potentials reveals that Cu2O is thermodynamically capable of photocatalytic reduction of CO2 to methanol and the reduction and oxidation of water.