The chemostat is an apparatus used by researchers for the continuous culture of microorganisms. A culture grown in a chemostat can be kept in a perpetual state of growth. The chemostat is very useful in problems concerning the formation of mutants by spontaneous mutation and the selection of different mutant strains. The objective of this proposal is to construct mathematical models for the development of a colony of microorganisms maintained in a chemostat. Particular attention will be given to stochastic models that provide a framework in which mutation experiments can be analyzed. Once the mathematical properties of the models are sufficiently understood, the models will be utilized in developing statistical procedures for use in conjunction with chemostats. Specifically, procedures will be developed for estimating mutation rates.