The intra-cluster and inter-galactic media that pervade the large scale structure of the Universe are known to be magnetized at sub-micro Gauss to micro Gauss levels and to contain cosmic rays. The acceleration of cosmic rays and their evolution along with that of magnetic fields in these media is still not well understood. Diffuse radio sources of synchrotron origin associated with the Intra-Cluster Medium (ICM) such as radio halos, relics and mini-halos are direct probes of the underlying mechanisms of cosmic ray acceleration. Observations with radio telescopes such as the Giant Metrewave Radio Telescope, the Very Large Array and the Westerbork Synthesis Radio Telescope have led to the discoveries of about 80 such sources and allowed detailed studies in the frequency range 0.15–1.4 GHz of a few. These studies have revealed scaling relations between the thermal and non-thermal properties of clusters and favour the role of shocks in the formation of radio relics and of turbulent re-acceleration in the formation of radio halos and mini-halos. The radio halos are known to occur in merging clusters and mini-halos are detected in about half of the cool-core clusters. Due to the limitations of current radio telescopes, low mass galaxy clusters and galaxy groups remain unexplored as they are expected to contain much weaker radio sources. Distinguishing between the primary and the secondary models of cosmic ray acceleration mechanisms requires spectral measurements over a wide range of radio frequencies and with high sensitivity. Simulations have also predicted weak diffuse radio sources associated with filaments connecting galaxy clusters. The Square Kilometre Array (SKA) is a next generation radio telescope that will operate in the frequency range of 0.05–20 GHz with unprecedented sensitivities and resolutions. The expected detection limits of SKA will reveal a few hundred to thousand new radio halos, relics and mini-halos providing the first large and comprehensive samples for their study. The wide frequency coverage along with sensitivity to extended structures will be able to constrain the cosmic ray acceleration mechanisms. The higher frequency (>5 GHz) observations will be able to use the Sunyaev–Zel’dovich effect to probe the ICM pressure in addition to tracers such as lobes of head–tail radio sources. The SKA also opens prospects to detect the ‘off-state’ or the lowest level of radio emission from the ICM predicted by the hadronic models and the turbulent re-acceleration models.