Quantum optics with free electrons examines coherent interactions between propagating electron wave packets and electromagnetic fields and material excitations. Unlike bound-electron systems, free electrons possess continuous energy spectra, yet they can exchange discrete photons via phase-matched interactions with optical modes in structured media, cavities, or near fields. This Review begins with a minimal physical picture of free electron–light interaction, then we develop a unified scattering framework that describes these interactions and explicitly discusses its semiclassical, effective singlemode, recoil-resolved, and fully multimode limits. Within this framework, we review the experiment and theoretical progress that demonstrate multiphoton energy ladders, spontaneous radiation, coherent electron-wave-packet shaping, electron–photon correlations, and conditional photonic-state preparation. We then discuss the proposals for electron-mediated quantum gates, error-correction primitives, quantum metrology, and quantum free-electron light sources, together with the experimental platforms and technical challenges relevant to fully quantum electron–light interfaces.