In the last 10 years, the improved capabilities (sensitivity, spectral and spatial resolution) of multi-wavelength telescopes have allowed us to study in detail the formation and evolution of the largest gravitationally bound systems in the Universe, i.e. galaxy clusters. Following the hierarchical scenario of structure formation, massive clusters form through episodic mergers of smaller mass units (groups and poor clusters) and through the continuous accretion of field galaxies.
It has now been proven that major cluster mergers, with their huge release of gravitational binding energy (~ 1064 ergs), deeply affect the physical properties of the different components of clusters, i.e. the temperature, metallicity and density distribution of the thermal intracluster medium (ICM) emitting in X-rays (e.g. Buote 2002, Sauvageot et al. 2005, Ferrari et al. 2006a, Kapferer et al. 2006, Markevitch & Vikhlinin 2007), the global dynamics and spatial distribution of galaxies (e.g. Girardi & Biviano 2002, Ferrari et al. 2003, Ferrari et al. 2005, Maurogordato et al. 2007), as well as their star-formation rate (e.g. Gavazzi et al. 2003, Poggianti et al. 2004, Ferrari et al. 2006b). The typical signatures that allow to identify merging clusters from optical and X-ray observations are: a) substructures in the X-ray and optical surface densities (see Buote 2002 and references therein), b) non-Gaussian radial velocity distributions of cluster members (see Girardi & Biviano 2002 and references therein), c) patchy ICM temperature, pressure, entropy and metallicity maps (e.g. Finoguenov et al. 2005, Kapferer et al. 2006), d) sharp X-ray surface brightness discontinuities, accompanied by jumps in gas and temperature ("cold fronts", see, e.g., Markevitch et al. 2000), e) absent or disturbed cooling-cores (e.g. Markevitch et al. 1999), f) larger core radii compared to (nearly) relaxed clusters (Jones & Forman 1999). There are also indications that recent merging events lead to a depletion of the nearest cluster neighbours (e.g. Schuecker & Böhringer 1999). Additionally, deep radio observations have revealed the presence of diffuse and extended (~ 1 Mpc) radio sources in about 50 merging clusters. Their radio emission is not related to a particular cluster member, but rather to the presence of relativistic electrons (Lorentz factor >> 1000) and weak magnetic fields (µG) in the intracluster space.
In this review, we focus on radio observations of this non-thermal component in galaxy clusters. We outline our current knowledge on the presence of non-thermal processes in the intracluster gas, and their physical connections with the thermodynamical evolution of large-scale structure. The relevance of the study of extended cluster radio emission for cosmology is pointed out. On smaller scales, there are only few indications of the possible presence of extended radio emission in galaxy groups. These systems host diffuse ~ 1 keV gas called intragroup medium (IGM) (see e.g., Mulchaey 1996). Radio and hard X-ray emission possibly related to a non-thermal component of the IGM has been recently pointed out by Delain & Rudnick (2006) and Nakazawa et al. (2007) respectively. The existence of diffuse radio sources in galaxy groups has indeed to be tested with observations of higher sensitivity. Radio observations of the emission from individual radio galaxies are not treated here. For a discussion on cluster radio galaxies see the reviews by Feretti & Venturi (2002) and Feretti & Giovannini (2007). X-ray observations and simulations of the non-thermal component in clusters are reviewed by Rephaeli et al. (2008) - Chapter 5, and Dolag et al. (2008) - Chapter 15, this volume. The adopted cosmology is CDM (H0 = 70 km s-1 Mpc-1, m = 0.3, = 0.7).