The theory of quantum electrodynamics has presented a complete calculation of the energy levels of positronium (Ps, e(+)e(-)) up to the order R(infinity)alpha(4). The annihilation rates for para-Ps (1(1)S(0)) and ortho-Ps (1(1)S(1)) are completely known up to the order R(infinity)alpha(5) and R(infinity)alpha(6) respectively. The present experimental uncertainty for spectroscopic results is 5-100 times larger than the theoretical uncertainty. For the annihilation rates the experimental situation is even worse. The theoretical predictions have presently reached a level of precision which is a challenge to experimentalists! The experimental uncertainty for the fine- and hyperfine transitions in the Ps atom is determined by the natural line width. An improvement is only possible with better statistics, i.e. higher positron intensities. For the annihilation rate of ortho-Ps there exist controversial measurements. Here completely new experiments are necessary which differ from the old measurements by possible systematic errors. All new proposals need high positron intensities which can be conveniently produced by bremsstrahlung and pair conversion in the beam dump of medium energy electron accelerators. (C) 2002 Elsevier Science B.V. All rights reserved.