Galactose oxidase is a mononuclear copper enzyme which oxidizes primary alcohols to aldehydes using molecular oxygen. A unique type of cross-link between tyrosine 272, an active site copper ligand, and cysteine 228 provides a modified tyrosine radical site which is believed to act as a one-electron redox center. Galactose oxidase is highly selective in its processing of hexopyranose substrates. Turnover of D-galactose is stereospecific for cleavage of the pro-S hydrogen. D-Galactose is an excellent substrate but its C-4 epimer D-glucose is not a substrate and will not even bind at 1 M concentration. Any proposed mechanism for galactose oxidase should be able to account for these stringent hexopyranose substrate specificities. In this paper we report molecular modeling studies of active site binding of postulated radical carbon-hydrogen bond cleavage transition states of D-galactose and D-glucose. Differences in specific enzyme-substrate interactions provide convincing explanations of the pro-S and galactose specificities. In addition, a previously unconsidered concerted radical mechanism appears to be just as plausible as the more standard stepwise radical mechanism via a ketyl radical anion intermediate. Regardless of whether a stepwise or concerted mechanism is operating, the active site appears to be well designed to bind radical transition states and perform radical enzyme catalysis. The detailed models developed here for ground state and transition state enzyme-substrate interactions provide insight to guide mechanistic studies using both radical-probing substrates and site-directed mutagenesis.