  The energy levels of the muonium ($\mu^+ e^-$) atom, which consists of two
''point-like'' leptonic particles, can be calculated to very high accuracy in
the framework of bound state Quantum Electrodynamics (QED), since there are no
complications due to internal nuclear structure and size which is the case for
all atoms and ions of natural isotopes including atomic hydrogen. The ground
state hyperfine splitting and the 1s-2s energy interval can provide both tests
of QED theory and determinations of fundamental constants like the muon mass
$m_{\mu}$ and the fine structure constant $\alpha$. The excellent understanding
of the electromagnetic binding in the muonium atom allows in particular testing
fundamental physical laws like lepton number conservation in searches for
muonium to antimuonium conversion and probing the nature of the muon as a heavy
leptonic object. New measurements on the ground state hyperfine splitting and
the 1s-2s transition are discussed.
