Bacterial populations have served as model systems for studying evolutionary processes ever since the classic experiments of Luria and Delbrück, which demonstrated the occurrence of mutations prior to selection for the traits they conferred. However, several authors have recently presented experiments suggesting that bacteria may have mechanisms for directing which mutations occur, such that the rate of adaptive mutations is enhanced. Before the hypothesis of directed mutation is accepted, it is imperative to consider alternative hypotheses that might account for the same observations. To this end, we expand upon existing mathematical theory of the dynamics of mutation and selection in clonal populations for two cases of particular interest. The first case concerns selection against mutants before plating; this selection occurs as the result of differences in growth rate between mutants and nonmutants. We demonstrate that this selection model gives rise to distributions of mutants, obtained by plating from sister cultures, that are very similar to those expected when some mutations are induced by the selective environment. The second case concerns the sequential incorporation of two mutations as the result of selection for an intermediate genotype after plating. We demonstrate that this two-step mutation model also yields distributions that are similar to those expected when some mutations are induced by the selective environment. These two cases therefore provide alternatives to the hypothesis of directed mutation. We suggest experiments that might be used to examine our alternative hypotheses. We also contrast the hypothesis of directed mutation with the notion of inheritance of acquired characteristics.