We report on the heteroepitaxial growth of high-quality single crystal cubic ZnxMg1-xO and NiyMg1-yO thin films by radio frequency oxygen plasma-assisted molecular beam epitaxy (RF-MBE). Film compositions over the ranges x = 0 to x = 0.65 and y = 0 to y = 1 have been grown on lattice-matched MgO (100) and characterized optically, morphologically, compositionally, and electrically. Both of these ternary materials are shown to have bandgaps which vary directly as a function of transition metal (Ni or Zn) concentration. Optical transmission measurements of NiyMg1-yO show the bandgap to shift continuously over the approximate range 3.5 eV (for NiO) to 4.81 eV (for y=0.075). Similarly, the bandgap of cubic ZnxMg1-xO is shifted from about 4.9 eV (for x = 0.65) to 6.25 eV (for x=0.12). Films exhibit good morphological quality and typical roughness of NiyMg1-yO films is 5 Å while that of ZnxMg1-xO is less than 15 Å, as measured by atomic force microscopy (AFM). X-ray diffraction (XRD) is employed to confirm crystal orientation and to determine the films' lattice constants. Film compositions are interrogated by Rutherford Backscattering (RBS) and electrical characterization is made by room-temperature Hall measurements.