Cuprous oxide (Cu2O) films were grown on polycrystalline and singlecrystalline copper substrates by plasma-assisted molecular beam epitaxy (PA-MBE) to evaluate the effects of substrate temperature, grain orientation, oxygen pressure and plasma power on the epitaxial growth of Cu2O. The copper substrate was annealed and electro-polished prior to the epitaxial growth. Following the MBE growth, the deposited films were analyzed by X-ray diffraction and electron backscatter diffraction equipped in a scanning electron microscope. The results indicate that Cu2O grows epitaxially on substrate grains with orientations distributed from <1 1 0>//ND to <1 1 1>//ND (here ND represent the normal direction of the sample) within a processing window of proper oxygen pressure and plasma power at 650 degrees C substrate temperature.