Yuliy Sannikov is an extraordinary theorist who has developed methods that offer new insights in analyzing problems that had seemed well-studied and familiar: for example, decisions that might bring about cooperation and/or defection in a repeated-play prisoner’s dilemma game, or that affect the balance of incentives and opportunism in a principal-agent relationship. His work has broken new ground in methodology, often through the application of stochastic calculus methods. The stochastic element means that his work naturally captures situations in which there is a random chance that monitoring, communication, or signaling between players is imperfect. Using calculus in the context of continuous-time games allows him to overcome tractability problems that had long hindered research in a number of areas. He has substantially altered the toolbox available for studying dynamic games. This essay offers an overview of Sannikov’s research in several areas.