Student understanding of the equilibrium coexistence of a liquid and its vapor was the subject of an extended investigation. Written assessment questions were administered to undergraduates enrolled in introductory physics and chemistry courses. Responses have been analyzed to document conceptual and reasoning difficulties in sufficient detail to be of practical use to instructors. Even after instruction on the relevant material, many students fail to recognize that for one-component systems in which a liquid and its vapor coexist in equilibrium, the pressure is controlled solely by the temperature. Although most students seem to realize that vaporization and condensation both take place, few are able to construct a coherent, step-by-step explanation for how dynamic phase equilibrium is established. Implications for instruction are discussed. (Contains 2 tables and 3 figures.)