Large-scale scientific computing is playing an ever-increasing role in critical decision-making and dynamic, event-driven systems. While some computation can simply wait in a job queue until resources become available, key decisions concerning life-threatening situations must be made quickly. A computation to predict the flow of airborne contaminants from a ruptured railcar must guide evacuation rapidly, before the results are meaningless. Although not as urgent, on-demand computing is often required to leverage a scientific opportunity. For example, a burst of data from seismometers could trigger urgent computation that then redirects instruments to focus data collection on specific regions, before the opportunity is lost. This paper describes the challenges of building an infrastructure to support urgent computing. We focus on three areas: the needs and requirements of an urgent computing system, a prototype urgent computing system called SPRUCE currently deployed on the TeraGrid, and future technologies and directions for urgent computing.