Zahavi (1975) proposed the handicap hypothesis for the adaptive evolution of female mating preferences and sexual selection that involves the joint evolution of a male secondary sexual character (the "handicap"), mating preferences for that character, and a third trait that affects the viability of individuals of both sexes. Under this hypothesis, when the degrees of expression of the handicap and viability traits are correlated, females that mate with the more handicapped males are selected for, since they bear offspring carrying genes for a high degree of expression of the viability trait. Several types of genetic models of the three characters evolving in a polygynous mating system show that the supposed adaptive evolution of mating preferences does not occur. Rather, there is an infinite continuum of possible evolutionary equilibria, and the female mating preference can equilibrate at any value. The point at which it does equilibrate determines the equilibrium degree of expression for the handicap and viability traits, which may be at points far from their ecological (natural-selection) optima. The qualitative result that female preferences do not consistently evolve in an adaptive manner holds under a variety of conditions, including (1) when the viability and handicap traits are under either stabilizing or directional natural selection, (2) when expression of the handicap is completely determined by nonheritable causes, and (3) when the handicap trait is expressed in both sexes rather than in males alone. These results can be understood in terms of a balance of forces in which the advantage that certain male phenotypes achieve through mating offsets their survival disadvantage. Thus, sexual selection, far from being adaptive, can in fact cause a serious reduction in the average survival rate of individuals in a population. The conclusion these analyses lead to is that in polygynous species in which males contribute only gametes (but not material benefits) to their mates, mating preferences may often evolve in unpredictable ways that do not enhance the average viability or adaptation of the species. For female mating preferences (and hence sexual selection) to evolve in an adaptive manner as suggested by the handicap hypothesis, some additional selective force operating directly on the mating preferences must be at work.