  Terms contributing to the hyperfine structure of the muonium ground state at
the level of few tenths of kHz have been evaluated. The $\alpha^2 (Z\alpha)$
radiative correction has been calculated numerically to the precision of 0.02
kHz. Leading $\ln (Z\alpha )$ terms of order $\alpha^{4-n} (Z\alpha)^n ,
n=1,2,3,$ and some relativistic corrections have been evaluated analytically.
The theoretical uncertainty is now reduced to 0.17 kHz. At present, however, it
is not possible to test QED to this precision because of the 1.34 kHz
uncertainty due to the muon mass.
