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Genetic models are presented for the analysis of quantitatively inherited characters which are expressed in the triploid endosperm of plants. These models are appropriate for various populations derived from crosses between two homozygous parents where, as a consequence, in the reference population the gene frequencies are half at all segregating loci. The first model, which allows for the estimation of genotypic values, including additive, dominance and first order epistatic effects, can be used with data from the two parental lines, two reciprocal F1 populations, the F2 and the four backcross populations using the F1 as one of the parents. The model was not extended to those four other backcrosses involving the FIR populations as one of the parents. A second model is presented for the estimation of genetic variance components, including the variances of first order epistatic effects. A third model is described for the estimation of genetic variance components in nested designs which arise in pedigree breeding programs using self fertilization. Data from appropriate experiments are used to illustrate the use of the models.