The growth of high-gap semiconductors (ΔEg>2 eV) on foreign substrates is discussed in terms of the properties of the semiconductor, the heterogeneous interface and the substrate. The creation of a heterogeneous interface is outlined and the need for analysis of reordering and other interface phenomena emphasized. Criteria adopted for the selection of substrates are: adhesion energy, chemical and thermal stability, crystal structure, surface properties, thermal expansion and availability. Semiconductors considered include GaN, AIN, ZnSe, SiC and diamond. Common substrates, such as sapphire, are shown to meet the criteria inadequately. All the criteria are important but the dominant one in use seems to be availability. It is suggested that the reason for the use of such an extremely small number of substrates during a decade of prolific activity in heteroepitaxy is that the cost of developing a commercial supply of any new crystal is too high to be supported by existing research programmes. It is proposed that heteroepitaxy is the universal technique in device development only because of the absence of good substrates of the semiconductors themselves. The availability of SiC substrates, for example, has removed the need for heteroepitaxy in SiC device development. Some unusual substrates, such as NdGaO3, Hf and ScAlMgO4 are reviewed briefly. © 1998 Chapman & Hall