Summary: Bands of motile Escherichia coli have been observed to travel at constant speed when the bacteria are placed in one end of a capillary tube containing oxygen and an energy source. Such bands are a consequence of a chemotatic mechanism which permits the bacteria to seek an optimal environment: the bacteria avoid low concentrations and move preferentially toward higher concentrations of some critical substrate. In this paper we develop a phenomenological theory of traveling bands starting with partial differential equations which describe the consumption of the critical substrate and the change in bacterial density due to random motion and to chemotaxis. The analysis shows that a band will form only if chemotaxis is sufficiently strong. The predicted band speed is shown to be in satisfactory agreement with observation. The analysis also predicts the shapes of the graphs of bacterial density and substrate concentration in the traveling band and shows how, from these shapes, one can determine a quantitative measure of the relative strength of chemotaxis.