We study epitaxial growth onto a patterned substrate by considering the nucleation and motion of steps on the surface when the surface is below its roughening temperature. The substrate is chosen as a periodic sequence of mesa structures composed of low-energy thigh-symmetry) facets and surfaces vicinal to these high-symmetry directions. Deposition to the surface, diffusion on terraces and adatom exchange with kink sites at the steps, nucleation on top of facets, particle migration from facets to vicinal surfaces and their incorporation into steps are treated explicitly. Repulsive and attractive interactions between step pairs are also taken into account Differential equations describing the motion of each individual step on the surface and the growth of facets normal to themselves are derived in accordance with the atomistic processes and step interactions mentioned above. Numerical integration of the final kinetic equations reveals a variety of growth morphologies depending upon the relative rates of these processes and step interactions. Step bunching due to attractive step interactions is observed for all cases considered. The strength of bunching decreases with the increase of the flux to the surface, but does not disappear completely. Scaling behavior of the time dependence of the height of the surface with respect to particle deposition rate to the surface is also observed. (C) 1999 Elsevier Science B.V. All rights reserved.