Non-thermal emission from galaxy clusters: a Pandora's vase for astrophysics

The existence of cosmic rays and weak magnetic fields in the intracluster volume has been well proven by deep radio observations of galaxy clusters. However a detailed physical characterization of the non-thermal component of large scale-structures, relevant for high-precision cosmology, is still missing. I will show the importance of combining numerical and theoretical works with cluster observations by a new-generation of radio, Gamma- and X-ray instruments.