The Compton scattering process plays a significant role in atomic, condensed matter, nuclear physics and material science. It could provide us useful information on the electromagnetic interaction between light and matter. Several aspects of many-body physics, such us electronic structures, electron momentum distributions, many-body interactions between bound electrons, and other properties, can be revealed by virtue of Compton scattering experiments. In this work, we give an review on ab initio calculation of Compton scattering process. We shall focus on several approachesfree electron approximation (FEA), impulse approximation (IA), incoherent scattering function / incoherent scattering factor (ISF) and scattering matrix (SM) -and discuss the main features as well as the available ranges for each approach. Furthermore, in this work, we also briefly introduce the databases and applications for Compton scattering.