Recent models of sexual selection suggest that females may prefer males that provide heritable benefits (`good genes') for offspring development or survival. We tested this possibility with a three-generation experiment using Trinidadian guppies, Poecilia reticulata, a species of livebearing freshwater fish. First, we show that female guppies were attracted to larger-bodied males. Areas of various colour pigments had no effect on female preference in this population. Second, male size had significant father-son heritability. Third, large fathers sired both sons and daughters with higher growth rates. Finally, the higher growth rates of daughters resulted in large reproductive output, attributable to their larger body size. Female mate preferences may therefore have important effects on the inheritance of life history traits by offspring. The results are consistent with the `good genes' theory of sexual selection but they also illustrate some of the pitfalls inherent in distinguishing among alternative theories.