Galaxy clusters are the largest storage rooms for cosmic material (galaxies, dark matter, hot thermal baryonic plasma, non-thermal and relativistic plasma, black holes, magnetic fields, cosmic rays). In this sense they can be considered as the largest multi-disciplinary laboratories in the universe where one can efficiently study some of the most interesting aspects of the astrophysics of large-scale structures: the nature of dark matter, the origin and distribution of cosmic rays, the impact of magnetic fields on large-scale structures, the impact of black holes on large-scale structures, the interplay between thermal and non-thermal phenomena.
As such, galaxy clusters are environments for the production of high-energy emission from the cluster diffuse medium. Many theoretical expectations have been produced but no clear evidence of high-E emission (at $E> 20$ keV) has been obtained so far.
We discuss in this paper the main results and controversies posed by theoretical and observational challenges on the physics of galaxy clusters by using a multi-frequency analysis.