The kinetics of four different reactions over two rare earth oxide catalysts have been determined at 923 K-N2O decomposition, NO decomposition, NO reduction by CH4 in the absence of O-2, and NO reduction by CH4 with excess O-2 present. After examination of many different reaction mechanisms and the rate expressions derived from each of them, the sequence of elementary steps and the resulting rate equation which best describe the experimental results for each are reported here. The Langmuir-Hinshelwood-type expressions indicate that N2O proceeds via unimolecular decomposition, whereas the rate determining step for NO decomposition involves a surface reaction between two adsorbed NO molecules. The reduction of NO by CH4 in the absence of O-2 proceeds with methyl radical formation via hydride abstraction by NO implied as the slow step, and the rate determining step for NO reduction when both CH4 and O-2 are present again appears to be methyl radical formation by reaction between adsorbed CH4 and NO2, The rate enhancements for NO reduction to N-2 and O-2 when CH4, or CH4 plus O-2, is utilized are attributed to the formation of methyl radicals and subsequent C-containing surface intermediates which rapidly interact with NO, The proposed chemistry is consistent with both that associated with methane oxidative coupling over rare earth oxides and that describing homogeneous free-radical reactions involving these intermediates. Consequently, a complete catalytic cycle of elementary steps is proposed for each reaction to demonstrate the possible presence of these various intermediates and to allow the testing of these models in the future.