  We investigate the spatial and spectral properties of non-thermal emission
from clusters of galaxies at gamma-ray energies between 10 keV and 10 TeV due
to inverse-Compton (IC) emission, pion-decay and non-thermal bremsstrahlung
(NTB) from cosmic-ray(CR) ions and electrons accelerated at cosmic shock and
secondary e+- from inelastic p-p collisions. We identify two main emission
region, namely the core (also bright in thermal X-ray) and the outskirts region
where accretion shocks occur. IC emission from shock accelerated CR electrons
dominate the emission in the outer regions of galaxy clusters, provided that at
least a fraction of a percent of the shock ram pressure is converted into CR
electrons. A clear detection of this component and of its spatial distribution
will allow us direct probing of cosmic accretion shocks. In the cluster core,
gamma-ray emission above 100 MeV is dominated by pion-decay mechanism and, at
lower energies, by IC emission from secondary e+-. However, IC emission from
shock accelerated electrons projected onto the cluster core will not be
negligible. We emphasize the importance of separating these emission components
for a correct interpretation of the experimental data and outline a strategy
for that purpose. Failure in addressing this issue will produce unsound
estimates of the intra-cluster magnetic field strength and CR ion content.
According to our estimate future space borne and ground based gamma-ray
facilities should be able to measure the whole nonthermal spectrum both in the
cluster core and at its outskirts. The importance of such measurements in
advancing our understanding of non-thermal processes in the intra-cluster
medium is discussed.
