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Abstract An extensive calculation of the optical-phonon-assisted transition rates for non-adiabatic electronic hopping motion in a solid is presented. Holstein's Molecular Crystal Model is used as a basis for study and the computation involves no restrictions on either the magnitude of the electron-lattice coupling strength, the temperature, or the difference between the electronic energies of the initial and final sites. In the strong-coupling small-polaron régime, the jump rates, associated d.c. conductivity, a.c. conductivity, and electric-field dependence of the d.c. conductivity, for a crystal are all calculated. These transport properties manifest qualitatively distinct behaviours corresponding to whether the temperature is above or well below the optical-phonon temperature. In the low-temperature régime the energy-conserving processes which involve the absorption of the minimum amount of vibrational energy provide the dominant contribution to the thermally activated jump rates. At sufficiently high temperatures multiphonon processes dominate the transition rate; the high-temperature jump rates are also activated, albeit with a different activation energy than that which characterizes the low-temperature régime. In the complementary weak-coupling régime, the jump rate is characterized by the dominance of those processes which involve the absorption or emission of the minimum number of phonons consistent with the requirements of energy conservation. Once again two distinct temperature domains manifest themselves: a low-temperature, thermally activated, régime and a high-temperature non-activated régime. Finally, it is shown that, while optical-phonon-assisted transition rates are well defined for a three-dimensional model, divergences will occur for a linear chain (at ω = 0) and for a planar model (at the optical phonon frequency). The calculations in this article bring together many of the results previously obtained for special cases into a single framework and reveal a number of new considerations. The article is the first of two, the second dealing with acoustic-phonon-assisted hopping.