To understand the effects of lattice mismatch on heteroepitaxial growth, we have studied the equilibrium structure and orientation relationships (ORs) of FCC films grown epitaxially on FCC substrates, using molecular dynamics simulations in conjunction with embedded atom method potentials. Three film/substrate systems have been investigated, namely: Ag on Cu, Ag on Ni and Pb on Al. These systems cover a significant range of lattice mismatch, from 12.6% for Ag/Cu to 21.8% for Pb/Al. For each system, the ORs of films on six different substrate orientations, namely: (100), (511), (311), (211), (322) and (111), have been investigated. Films on these susbstrates cover a gradual transition from the oct-cube orientation relationship, which occurs only on (100) substrates, to the heterotwin orientation relationship, which often occurs on (111) substrates. It is found that the resulting ORs vary systematically with substrate orientation, but that the pattern of variation is almost identical for all three systems, and therefore largely independent of mismatch. However, the manner in which mismatch is accommodated does depend on the magnitude of mismatch. Simulations point to an important role for edge-to-edge matching and defects such as stacking faults. An analysis of these results in terms of transformation strains highlights the distinction between the ORs, which are largely independent of mismatch, and the local interfacial structure, which changes directly with mismatch.