Theoretical and experimental methods are applied to the study of the reaction dynamics in hyperthermal collisions of O+ (S-4(3/2)) with methane. Measurements of the absolute reaction cross sections for the interaction of O+ with CD4 and CH4 were obtained at collision energies in the range from near-thermal to approximately 15 eV, using the guided-ion beam (GIB) technique. Product recoil velocity distributions, using the GIB time-of-flight (TOF) methods, were determined for several product ions at selected collision energies. The main reaction channel, charge transfer, proceeds via large impact parameter collisions. A number of minor channels, involving more intimate collisions, were also detected. Ab initio electronic structure calculations have been performed with different levels of theory and basis sets, including high-level coupled-cluster calculations to determine the energies of reaction intermediates and transition states for reaction. Several reaction paths on both quartet and doublet electronic states of (O.CH4)(+) are found, and these provide a reasonable qualitative interpretation of the experiments. Although most of the products can be produced via spin-allowed pathways, the appearance of CH3+ at low energies suggests that intersystem crossing plays some role.