Abstract Positron sources are the key elements for the future and current lepton collider projects such as ILC, CLIC, SuperKEKB, FCC-ee, Muon Collider/LEMMA, etc., introducing challenging critical requirements for high intensity and low emittance beams in order to achieve high luminosity. In fact, due to their large production emittance and constraints given by the target thermal load, the main collider parameters such as the peak and average current, the emittances, the damping time, the repetition frequency and consequently the luminosity are determined by the positron beam characteristics. In this paper, the conventional positron sources and their main properties are explored for giving an indication to the challenges that apply during the design of the advanced accelerator concepts. The photon-driven positron sources as the novel approach proposed, primarily for the future linear colliders, are described highlighting their variety and problematic.