Cyber threat hunting is the process of proactively and iteratively formulating and validating threat hypotheses based on security-relevant observations and domain knowledge. To facilitate threat hunting tasks, this paper introduces threat intelligence computing as a new methodology that models threat discovery as a graph computation problem. It enables efficient programming for solving threat discovery problems, equipping threat hunters with a suite of potent new tools for agile codifications of threat hypotheses, automated evidence mining, and interactive data inspection capabilities. A concrete realization of a threat intelligence computing platform is presented through the design and implementation of a domain-specific graph language with interactive visualization support and a distributed graph database. The platform was evaluated in a two-week DARPA competition for threat detection on a test bed comprising a wide variety of systems monitored in real time. During this period, sub-billion records were produced, streamed, and analyzed, dozens of threat hunting tasks were dynamically planned and programmed, and attack campaigns with diverse malicious intent were discovered. The platform exhibited strong detection and analytics capabilities coupled with high efficiency, resulting in a leadership position in the competition. Additional evaluations on comprehensive policy reasoning are outlined to demonstrate the versatility of the platform and the expressiveness of the language.