Mathematical models of the transmission dynamics of scrapie are used to explore the expected course of an outbreak in a sheep flock, and the potential impacts of different control measures. All models incorporate sheep demography, a long and variable scrapie incubation period, horizontal and vertical routes of transmission and genetic variation in susceptibility. Outputs are compared for models which do and do not incorporate an environmental reservoir of infectivity, and which do and do not incorporate carrier genotypes. Numerical analyses using parameter values consistent with available data indicate that, in a closed flock, scrapie outbreaks may have a duration of several decades, reduce the frequency of susceptible genotypes, and may become endemic if carrier genotypes are present. In an open flock, endemic scrapie is possible even in the absence of carriers. Control measures currently or likely to become available may reduce the incidence of cases but may be fully effective only over a period of several years.