  We investigate thermal synchrotron radiation at semi-relativistic and
relativistic temperatures. We find an analytic expression for the
angle-averaged emission coefficient, and show that it is significantly more
accurate that those derived previously. We also present analytic approximations
to the synchrotron turnover frequency. Then, we treat Comptonization of the
synchrotron radiation, and give simple expressions for the spectral shape and
the emitted power. We also consider modifications of the above results by
bremsstrahlung. The importance of Comptonization of synchrotron radiation in
compact X-ray sources is then studied. We consider emission from hot accretion
flows and from active coronae above optically-thick accretion discs in
black-hole binaries and AGNs. Synchrotron Comptonization is found to be, in
general, negligible in luminous sources, except for those with hardest X-ray
spectra and stellar masses. Increasing the black-hole mass results in a strong
reduction of the maximum Eddington ratio possible from to this process. X-ray
spectra of intermediate-luminosity sources, e.g., low-luminosity AGNs, can be
explained by synchrotron Comptonization only in the case of hot accretion
flows. Then, bremsstrahlung emission always dominates X-ray spectra of very
weak sources. Finally, we consider weakly-magnetized neutron stars and find
that synchrotron Comptonization can account for the power-law X-ray spectra
observed in their low states.
