Difficulties caused by hadronic size effects encountered in quantum electrodynamics (QED) calculations of Lamb shifts in the hydrogen atom (H) are missing in positronium (Ps), a purely leptonic system. Precision measurements of the hyperfine structure in Ps thus provide less ambiguous tests of QED. A review of this problem, leading up to present-generation two-photon absorption experiments, is presented. For future investigations of Ps with two-colour lasers, calculations of the 13S1-23S1and 13S1-33S1two-photon transition amplitudes and probabilities are carried out. A new feature is the occurrence of a ‘destructive interference profile’ (DIP) in then= 3 pre-resonance region.