For a long time the necessity of having intense positron beams to get a high rate of events with fixed targets as with e+e− collisions, led to conventional positron sources using high energy and high intensity electron beams generating bremsstrahlung photons in thick high Z targets. The inconveniences for such scheme, heat load and large emittance, are underlined. Alternative ways aiming to provide a high number of photons impinging on rather thin conversion targets have been developed: they are using magnetic undulators, laser Compton backscattering and channeling radiation in axially oriented crystals. A brief description of these methods is given. However, the emphasis is put here on the channeling radiation. The advantages of channeling radiation, described by M.Kumakhov and others, were confirmed by many simulations and experiments. We are reporting, hereafter, on two schemes: positron all-crystal converters involving thick crystals where photon radiation and e+e− pair creation are occurring in the crystal and hybrid sources where the two functions are separated. With the hybrid scheme, including a thin crystal-radiator and a thick amorphous-converter, the heat load and the local deposited energy density can be lowered. Simulations and experiments related to these two schemes are reported. Further developments concerning the converter, granular instead of bulk, are providing additional advantages. The promising, already gathered results led to consider these kinds of positron sources for future e-e+ colliders. More physicists, from different laboratories, concerned by the studies on positron sources are now contributing in this field.