Thermal emission is the radiation of electromagnetic waves from hot objects. The promise of thermal-emission engineering for applications in energy harvesting, radiative cooling, and thermal camouflage has recently led to renewed research interest in this topic. There is a substantial need for accurate and precise measurement of thermal emission in a laboratory setting, which can be challenging in part due to the presence of background emission from the surrounding environment and the measurement instrument itself. This is especially true for measurements of emitters at temperatures close to that of the environment, where the impact of background emission is relatively large. In this paper, we describe, recommend, and demonstrate general procedures for thermal-emission measurements that are applicable to most experimental conditions, including less-common and more-challenging cases that include thermal emitters with temperature-dependent emissivity and emitters that are not in thermal equilibrium.