In populations of species such as cattle and sheep which are being selected for commercially important quantitative traits like milk yield and growth rate, the breeder cannot avoid overlapping generations since the female reproductive rate is so low. There is a classical theory for predicting the rate of response to selection in continuing programmes in which the selection differentials and age structure of the population do not change (Rendel and Robertson, (1950)). In a new programme, however, these rates of response are approached only asymptotically and may not be a useful approximation for several years. Recurrence relations have been developed by several authors to enable predictions of short term responses, but these can be simplified with Leslie's (1945) matrix formulation. Some basic principles will be given here; for details see Hill (1974).