Rate coefficients for inelastic Mg+H collisions are calculated for all transitions between the lowest seven levels and the ionic state (charge transfer), namely Mg(3s(2) S-1, 3s3p P-3, 3s3p P-1, 3s4s S-3, 3s4s S-1, 3s3d D-1, 3s4p P-3)+H(1s) and Mg+(3s S-2)+H-. The rate coefficients are based on cross-sections from full quantum scattering calculations, which are themselves based on detailed quantum chemical calculations for the MgH molecule. The data are needed for non-LTE applications in cool astrophysical environments, especially cool stellar atmospheres, and are presented for a temperature range of 500-8000 K. From consideration of the sensitivity of the cross-sections to various uncertainties in the calculations, most importantly input quantum chemical data and the numerical accuracy of the scattering calculations, a measure of the possible uncertainties in the rate coefficients is estimated.