Both DFT and G4 molecular orbital calculations have been employed to reexamine the mechanism of dimethyldioxirane (DMDO) oxidation of saturated hydrocarbons and the epoxidation of alkenes. The UM062X DFT functional provided the most accurate bond O-O dissociation energies for a series of typical peroxides. A diradicaloid process initiated by an O-O homolytic bond cleavage involving abstraction of hydrogen from the C-H bond followed by a final product forming "oxygen rebound" step best describes the DMDO oxidation of saturated hydrocarbons. In contrast, this study showed that the DMDO epoxidation of alkenes is a concerted process best described with the B3LYP DFT functional.