Positronium (Ps) is an exotic hydrogenic atom composed of an electron bound to a positron via the Coulomb force. Being composed of two low-mass leptons, positronium is, for all practical purposes, fully described by quantum electrodynamics (QED). The absence of hadronic components suggests that positronium energy levels and decay rates can be calculated to very high precision, limited only by the order of the corresponding perturbative expansion and the tiny effects of heavy or weakly interacting virtual particles and exotic decay modes. Moreover, as it is a low-mass particle–antiparticle system, the QED description of positronium is strongly affected by annihilation and recoil effects that are either weaker or not present in other atoms. As a result, sufficiently precise measurements of Ps energy levels and decay properties can serve as stringent tests of bound-state QED theory, and may be sensitive to processes not present in the theory, such as axion-like particles (beyond the QCD axion), or a fifth fundamental force. In addition, since positronium is an eigenstate of the fundamental symmetries C and P, various symmetry violating mechanisms can be probed through searches for anomalous decay modes. In the last three decades, there have been significant experimental advances in positron and positronium physics which open up the possibility to test QED bound-state theory with unprecedented precision. Here we present the current state-of-the-art in experimental positronium spectroscopy, and discuss explicitly how such measurements can be used to test bound-state QED theory, and how such tests may contribute to the search for physics beyond the Standard Model.