  The hydrogen-like muonium atom ($\mu^+e^-$) consists of a positive muon
($\mu^+$) and an electron ($e^-$). Since it was first observed by Hughes et al.
in 1960, a series of precision experiments could be carried out testing bound
state Quantum Electrodynamics (QED) and the nature of the muon as a point-like
leptonic particle. Precision values for fundamental constants could be
obtained, including the fine structure constant $\alpha$ and the muon's
magnetic moment $\mu_{\mu}$. Since no internal structure could be revealed for
leptons down to dimensions of $10^{-18}$m, the accuracy of theoretical
calculations of term energies is higher than for the natural hydrogen atom and
its isotopes deuterium and tritium as well as exotic hydrogen-like systems
which contain hadrons. Internal structure effects of these particles spoil the
precision of theoretical results. Of particular interest for precision muonium
experiments, which are carried out in order to test standard theory and to
extract accurate values for fundamental constants, are the ground state
hyperfine structure splitting $\Delta \nu_{HFS}$ and the 1s-2s interval $\Delta
\nu_{1s2s}$. Both cases involve the n=1 ground state in which the atoms can be
produced in sufficient quantities.
