Compton scattering is of prime importance to radiobiology, as it happens to be the most probable interaction of high energy X rays with atomic nuclei in living beings and is applied in radiation therapy.
In material physics, Compton scattering can be used to probe the wave function of the electrons in matter in the momentum representation.
Compton Scatter is an important effect in Gamma spectroscopy which gives rise to the Compton edge, as it is possible for the gamma rays to scatter out of the detectors used. Compton suppression is used to detect stray scatter gamma rays to counteract this effect.
In radiation health physics, Compton Scattering is used in some Monte Carlo codes that are used to evaluate the effectiveness of radiation shielding. Essentially, the code calculates the scattering angles of radiation that interacts with the shield, and depending on the angle, this contributes to the particle flux on the other side of the shield, which contributes to teh exposure rate.