Acceleration of cosmic-ray electrons (CRe) in the intra-cluster medium (ICM) is probed by radio observations that detect diffuse, megaparsec-scale, synchrotron sources in a fraction of galaxy clusters. Giant radio halos are the most spectacular manifestations of non-thermal activity in the ICM and are currently explained assuming that turbulence, driven during massive cluster–cluster mergers, reaccelerates CRe at several giga-electron volts. This scenario implies a hierarchy of complex mechanisms in the ICM that drain energy from large scales into electromagnetic fluctuations in the plasma and collisionless mechanisms of particle acceleration at much smaller scales. In this paper we focus on the physics of acceleration by compressible turbulence. The spectrum and damping mechanisms of the electromagnetic fluctuations, and the mean free path (mfp) of CRe, are the most relevant ingredients that determine the efficiency of acceleration. These ingredients in the ICM are, however, poorly known, and we show that calculations of turbulent acceleration are also sensitive to these uncertainties. On the other hand this fact implies that the non-thermal properties of galaxy clusters probe the complex microphysics and the weakly collisional nature of the ICM.