Thermal emission is a ubiquitous and fundamental process by which all objects at non-zero temperatures radiate electromagnetic energy. This process is often assumed to be incoherent in both space and time, resulting in broadband, omnidirectional light emission toward the far field, with a spectral density related to the emitter temperature by Planck's law. Over the past two decades, there has been considerable progress in engineering the spectrum, directionality, polarization and temporal response of thermally emitted light using nanostructured materials. This Review summarizes the basic physics of thermal emission, lays out various nanophotonic approaches to engineer thermal emission in the far field, and highlights several applications, including energy harvesting, lighting and radiative cooling.