We report on the growth of cuprous oxide on MgO(001) by plasma-assisted molecular beam epitaxy. An increased copper to oxygen ratio enhances the cube-on-cube growth of the Cu2O crystallites compared to the Cu2O(110)parallel to MgO(100) oriented crystallites. This correlates with the occurrence of a (3 root 2x root 2)R45 degrees surface reconstruction observed during growth. This reconstruction is known to be unstable under oxygen-rich conditions and stabilizes the Cu2O(001) surface compared to the Cu2O(110) surface for higher copper to oxygen ratios enhancing cube-on-cube growth. High resolution X-ray diffraction analysis of the resulting cube-on-cube Cu2O layers indicates partial relaxation by 39.7%. Investigations of the optical absorption and photoluminescence properties of the films reveal excitonic contributions comparable to those of single crystals. (C) 2015 Elsevier B.V. All rights reserved.