  The Compton scattering process plays significant roles in atomic and
molecular physics, condensed matter physics, nuclear physics and material
science. It could provide 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
of bound electrons, \emph{etc}, can be revealed by Compton scattering
experiments. In this work, we give a review on \emph{ab initio} calculation of
Compton scattering process. Several approaches, including the free electron
approximation (FEA), impulse approximation (IA), incoherent scattering function
/ incoherent scattering factor (ISF) and scattering matrix (SM) are focused in
this work. The main features and available ranges for these approaches are
discussed. Furthermore, we also briefly introduce the databases and
applications for Compton scattering.
