Two-factor mating designs at consecutive Sn and S(n+1) levels (S0 and S1 S1 and S2, or F2 and F3) allow estimation of all components of the variation among homozygous lines and F1 hybrids that can be derived from a given population. They also allow for the prediction of the mean of these lines and single-cross hybrids. Some tests for the presence of epistasis are possible at the levels of means and of variances. Such mating designs can be very useful for predicting the value of the best possible lines or the best possible F1 hybrids when it is difficult to produce, at an experimental level for exploratory purposes, either lines or hybrids.