Hospital diagnostic facilities, such as magnetic resonance imaging centers, typically provide service to several diverse patient groups: outpatients, who are scheduled in advance; inpatients, whose demands are generated randomly during the day; and emergency patients, who must be served as soon as possible. Our analysis focuses on two interrelated tasks: designing the outpatient appointment schedule, and establishing dynamic priority rules for admitting patients into service. We formulate the problem of managing patient demand for diagnostic service as a finite-horizon dynamic program and identify properties of the optimal policies. Using empirical data from a major urban hospital, we conduct numerical studies to develop insights into the sensitivity of the optimal policies to the various cost and probability parameters and to evaluate the performance of several heuristic rules for appointment acceptance and patient scheduling.