Abstract Relativistic positronium production is a theoretical development started by me [Olsen, H.A., 1986. Phys. Rev. D 33, 2033.] as an attempt to obtain beams of singlet and triplet positronium. In this way positronium experiments can be performed as in particle physics, which means that positronium is observed in vacuum and experiments are performed under well-controlled circumstances. The development of the theory is performed via the Born approximation of photo- and electro-production, and via the by now well-known high energy Coulomb corrected pair production and bremsstrahlung calculations [Olsen, H.A., Maximon, L.C., 1959. Phys. Rev. 114, 887]. In this way the photo- and electro-production mechanisms are obtained rather directly. Cross sections for positronium production and possible transfer of polarization from initial particle, electron or photon, to final positronium is obtained.