Pure Cu films were deposited on (0001) alpha-Al2O3 by molecular beam epitaxy at 473K to study the growth behaviour, orientation relationship and interface structure. In contrast, Cu/Ti bilayers, consisting of an 10nm thick Ti interfacial layer between the Cu film and (0001) alpha-Al2O3 substrate were grown at 373K. In situ reflection high-energy electron diffraction (RHEED) and transmission electron microscopy (TEM) results show that Cu/Ti layered films grow two-dimensionally on (0001) Al2O3 leading to an epitaxial orientation relationship between Cu, Ti and alpha-Al2O3: (111)(Cu) <110>(Cu) parallel to (0001)(Ti) [2 (1) over bar (1) over bar 0](Ti) parallel to (0001)(Al2O3) [10 (1) over bar 0](Al2O3). In the case of pure Cu films on alpha-Al2O3 the RHEED and TEM observations indicate that the epitaxy evolves in two stages. In the first stage the RHEED patterns reveal random orientation of the three-dimensional Cu nuclei. Then, in the second stage these polycrystalline crystallites coarsen and seek a lower energy orientation with the substrate. This reorientation leads to epitaxial Cu films on alpha-Al2O3 at 473K. TEM confirms that Cu layers on alpha-Al2O3 are epitaxial exhibiting an (111)(Cu) <110>(Cu) parallel to (0001)(Al2O3) [10 (1) over bar 0](Al2O3) orientation relationship.