We report fundamental issues and prospects of the rhombohedral corundum-structured III-oxide (III-O) alloy system constituted with alpha-M2O3 (M=metal element) materials. Successful epitaxial growth of alpha-Ga2O3 and alpha-In2O3 on sapphire (alpha-Al2O3) substrates has enabled the growth of alpha-(Al,Ga,In)(2)O-3 semiconductor alloys, achieving the "band gap engineering" from 3.8 eV (an experimental value) to 8.8 eV. Transition-metal corundum-structured oxides (M=Cr, Fe, V, Ti) can be alloyed with the alpha-(Al,Ga,In)(2)O-3 semiconductor alloys, leading to "function engineering", that is, to tailor the materials functions. The corundum-structured III-O alloys will contribute to novel multifunctional devices in the future. (C) 2014 Elsevier B.V. All rights reserved.