Most thermal radiation problems are analyzed in a “forward” manner, in which the aim is to predict the response of a system based on well-defined boundary conditions. In practice, however, many thermal radiation problems are inverse problems. For example, the goal of many furnace design problems is to find a configuration that realizes a particular irradiation profile on a target, while in measurement problems, transmitted or reflected radiation measured with sensors at a boundary may be used to infer the properties of matter within the boundary. Such inverse problems are often mathematically ill-posed because they may have multiple solutions or no solution at all. Consequently, analyzing these types of problems is more complex than is required for forward problems. In this review, we examine the various types and characteristics of inverse problems, outline standard inverse solution methods for them, and review the historical and contemporary literature.