In Fisher's model of sexual selection female mating preferences are not subject to direct selection but evolve purely because they are genetically correlated with the favoured male trait. But when female choice is costly relative to random mating, for example in energy, time or predation risks, the evolution of female mating preference is subject also to direct selection. With costly female choice the set or line of equilibria found in models of Fisher's process no longer exists. On the line the male trait is under zero net selection, and there is no advantage for a female choosing a male with a more exaggerated character. Therefore any cost to choice causes choosiness to decline. In turn this lowers the strength of sexual selection and the male trait declines as well. So when Fisher's process is the sole force of sexual selection and female choice is costly, only transitory increases in female choice and the preferred male trait are possible. It has often been claimed that exaggerated male characters act as markers or revealers of the genetic quality of potential mates. If females choose their mates using traits that correlate with heritable viability differences then stable exaggeration of both female choice and the preferred male character is possible, even when female choice is costly. The offspring of choosy females have not only a Fisherian reproductive advantage but also greater viability. This suggests that in species with exaggerated male ornamentation, in which female choice is costly, it is likely that female mate choice will be for traits that correlate with male genetic quality.