  Recent observations of high energy (> 20 keV) X-ray emission in a few
clusters of galaxies broaden our knowledge of physical phenomena in the
intracluster space. This emission is likely to be nonthermal, probably
resulting from Compton scattering of relativistic electrons by the cosmic
microwave background (CMB) radiation. Direct evidence for the presence of
relativistic electrons in some 50 clusters comes from measurements of extended
radio emission in their central regions. We briefly review the main results
from observations of extended regions of radio emission, and Faraday rotation
measurements of background and cluster radio sources. The main focus of the
review are searches for nonthermal X-ray emission conducted with past and
currently operating satellites, which yielded appreciable evidence for
nonthermal emission components in the spectra of a few clusters. This evidence
is clearly not unequivocal, due to substantial observational and systematic
uncertainties, in addition to virtually complete lack of spatial information.
If indeed the emission has its origin in Compton scattering of relativistic
electrons by the CMB, then the mean magnetic field strength and density of
relativistic electrons in the cluster can be directly determined. Knowledge of
these basic nonthermal quantities is valuable for the detailed description of
processes in intracluster gas and for the origin of magnetic fields.
