  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 \emph{ab initio} calculation of Compton
scattering process. We shall focus on several approaches -- free 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.
