Managing scarce resources plays a significant role in hospital operations. Effective use of resources (e.g., operating rooms, specialized doctors, etc.) allows hospitals to efficiently provide high-quality care to patients. In this dissertation, we study how hospitals manage scarce resources to identify systematic ways in which quality of care and efficiency might be improved. We study four different types of hospital resources: post-operative beds, specialist surgeons, resident physicians, and patient information. For each resource type, we show how better utilization could increase the quality of care delivered by the hospital or increase the efficiency of the system. We show that as post-operative bed utilization increases the discharge rate increases as well, meaning that bed shortages impact physician decision making. Further, we show that patients discharged on days with high bed utilization are significantly more likely to be readmitted to the hospital within 72 hours, which implies that poor bed management can lead to worse health outcomes for surgical patients. We also study how quality of care differs between night and day arrival in trauma centers. Based on a large national dataset, we conclude that a lack of specialized resources at hospitals during the off hours leads to significantly worse patient outcomes, including higher mortality and longer lengths of stay. Further, we exploit a natural experiment to determine the impact that residents have on efficiency in an academic emergency department. Using regression analysis, queueing models, and simulation, we find that when residents are present in the emergency department, treatment times are lowered significantly, especially among high severity patients. Finally, we show two novel uses of medical data to predict patient outcomes. We develop models to predict which patients will require an ICU bed after being transferred from outside hospitals to an internal medicine unit, using only five commonly measured medical characteristics of the patient. We also develop a model using MRI data to classify whether or not prostate cancer is present in an image.