The oxidation of methane with molecular oxygen using the atomic platinum cation as a catalyst, yielding methanol, formaldehyde, and higher oxidation products, has been studied both computationally and experimentally. The most relevant reaction pathways have been followed in detail. To this end a large number of stationary points, both minima and transition states, have beet? optimized using a hybrid density functional theory method (B3LYP). At these optimized geometries, energies have been calculated using both an empirical scaling scheme (PCI-80) and the B3LYP method employing extended basis sets with several polarization functions. Good agreement with available experimental data has been obtained. For the parts of the catalytic cycle where detailed experimental results have not been available, the new calculated results have complemented the experimental picture to reach an almost complete understanding of the reaction mechanisms. Spin-orbit effects have been incorporated using an empirical approach, which has lead to improved agreement with experiments. The new FTICR experiments reported in the present study have helped to clarify some of the most complicated reaction sequences.