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When impinging on optical structures or passing in their vicinity, free electrons can spontaneously emit electromagnetic radiation, a phenomenon generally known as cathodoluminescence. Free-electron radiation comes in many guises: Cherenkov, transition, and Smith-Purcell radiation, but also electron scintillation, commonly referred to as incoherent cathodoluminescence. While those effects have been at the heart of many fundamental discoveries and technological developments in high-energy physics in the past century, their recent demonstration in photonic and nanophotonic systems has attracted a lot of attention. Those developments arose from predictions that exploit nanophotonics for novel radiation regimes, now becoming accessible thanks to advances in nanofabrication. In general, the proper design of nanophotonic structures can enable shaping, control, and enhancement of free-electron radiation, for any of the above-mentioned effects. Free-electron radiation in nanophotonics opens the way to promising applications, such as widely-tunable integrated light sources from x-ray to THz frequencies, miniaturized particle accelerators, and highly sensitive high-energy particle detectors. Here, we review the emerging field of free-electron radiation in nanophotonics. We first present a general, unified framework to describe free-electron light-matter interaction in arbitrary nanophotonic systems. We then show how this framework sheds light on the physical underpinnings of many methods in the field used to control and enhance free-electron radiation. Namely, the framework points to the central role played by the photonic eigenmodes in controlling the output properties of free-electron radiation (e.g., frequency, directionality, and polarization). [... see full abstract in paper]