Air-breathing electric propulsion (ABEP) allows for lowering the altitude of spacecraft operations below 250 km, in the so-called Very Low Earth Orbits (VLEOs). Operations in VLEOs will give radical advantages in terms of orbit accessibility, payload performance, protection from radiations, and end-of-life disposal. ABEP combines an intake to collect the residual atmosphere in front of the spacecraft and an electric thruster to ionize and accelerate the atmospheric particles. Such residual gas can be exploited as a renewable resource not only to keep the spacecraft on a VLEO, but also to remove the main limiting factor of spacecraft lifetime, i.e., the amount of stored propellant. Several realizations of the ABEP concept have been proposed, but the few end-to-end experimental campaigns highlighted the need to improve the concept functional design and the representativeness of simulated atmospheric flows. The difficulty in recreating the VLEO environment in a laboratory limits the data available to validate scaling laws and modelling efforts. This paper presents a comprehensive review of the main research and development efforts on the ABEP technology.