  There is a guaranteed background of stochastic gravitational waves produced
in the thermal plasma in the early universe. Its energy density per logarithmic
frequency interval scales with the maximum temperature $T_{\rm max}$ which the
primordial plasma attained at the beginning of the standard hot big bang era.
It peaks in the microwave range, at around $80\,{\rm
GHz}\,[106.75/g_{*s}(T_{\rm max})]^{1/3}$, where $g_{*s}(T_{\rm max})$ is the
effective number of entropy degrees of freedom in the primordial plasma at
$T_{\rm max}$. We present a state-of-the-art prediction of this Cosmic
Gravitational Microwave Background (CGMB) for general models, and carry out
calculations for the case of the Standard Model (SM) as well as for several of
its extensions. On the side of minimal extensions we consider the Neutrino
Minimal SM ($\nu$MSM) and the SM - Axion - Seesaw - Higgs portal inflation
model (SMASH), which provide a complete and consistent cosmological history
including inflation. As an example of a non-minimal extension of the SM we
consider the Minimal Supersymmetric Standard Model (MSSM). Furthermore, we
discuss the current upper limits and the prospects to detect the CGMB in
laboratory experiments and thus measure the maximum temperature and the
effective number of degrees of freedom at the beginning of the hot big bang.
