New accurate theoretical rate coefficients for (de)-excitation and charge transfer in low-energy O + H, O+ + H− and O− + H+ collisions are reported. The calculations of cross-sections and rate coefficients are performed by means of the quantum probability current method, using full configuration interaction ab initio electronic structure calculations that provide a global description of all 43 lowest molecular states from short to asymptotic internuclear distances. Thus, both long- and short-range non-adiabatic regions are taken into account for the first time. All the doublet, quartet and sextet OH molecular states, with excitation energy asymptotes up to 12.07 eV, as well as the two lowest ionic states with the asymptotes O−H+ and O+H− are treated. Calculations are performed for the collision energy range 0.01–100eV and the temperature range 1 000–10 000 K. The mechanisms underlying the processes are analysed: it is shown that the largest rate coefficients, with values exceeding 10−8 cm3 s−1, are due to ionic–covalent interactions present at large internuclear distances, while short-range interactions play an important role for rates with moderate values involved in (de)-excitation processes. As a consequence, a comparison of the present data with previously published results shows that differences of up to several orders of magnitude exist for rate coefficients with moderate values. It is worth pointing out the relatively large rate coefficients for triplet–quintuplet oxygen transitions, as well as for transitions between the O(2p33s5So) and O(2p33p5P) levels of the oxygen triplet and H(n = 2) levels. The calculated data are important for modelling stellar spectra, leading to accurate oxygen abundances.