Epitaxial Cu films were grown on H-terminated Si(100), Si(110) and Si(111) substrates by magnetron sputtering. The epitaxial orientation relationships and microstructural characteristics of the Cu films were studied by x-ray diffraction (XRD) including the conventional theta-2 theta mode, rocking curve and pole figures, as well as by transmission electron microscopy. The results of both pole figure and electron diffraction reveal the epitaxial orientation relationship of the Cu/Si epitaxial system is as follows : Cu(100)/Si(100) with Cu[010]//Si[011]; Cu(111)//Si(110) with Cu[(1) over bar 10]//Si[001] and Cu[110]//Si[001] which are twin related; and for the Cu/Si(111) system the Cu film grows primarily in the epitaxial relationship of Cu(111)/Si(111) with Cu[1 (1) over bar 0]//Si[(2) over bar 11]. It is shown by XRD that Si(110) is a more favorable substrate than Si(111) for the epitaxial growth of Cu(111). An ultrathin Cu(111) film (up to 2.5 nm) with high epitaxial quality can be grown on Si(110). The epitaxial relationships of the Cu/Si are discussed on the basis of geometrical lattice matching, including the invariant-line criterion and the superlattice area mismatch rule.