The realization that heterosis is due chiefly to the dispersion of dominant genes between the parents of a cross rather than to the overdominance of these genes has led to the important practical conclusions that, in principle, heterosis should be fixable in inbred lines. It follows, therefore, that there is no longer any genetical justification for regarding the improvement of outbreeding species any differently from that of inbreeding ones, so that all crops and stock can be improved by inbreeding programmes. In the past, breeders have had to employ an essentially empirical approach in carrying out improvement programmes which involve inbreeding. Recent work, however, has led to an understanding of the theoretical basis of the advances to be expected under inbreeding, both with and without selection, so that it is now possible for the breeder to design and carry out improvement programmes in a systematic way and, hence, to predict, from the early generations of a pedigree, the likelihood of obtaining recombinant inbred lines or hybrids that have the desired level of performance. In this paper we review this theory and discuss its application to the practical task of crop improvement.