Compton scattering is a significant physical phenomenon, arising from the interaction between electrons and the electromagnetic field. In this paper, the wavelength shift of photons scattered by relativistically moving electrons is calculated using classical quantum theory and quantum electrodynamics (QED). Taking X-rays of a specific wavelength as an example, this paper employs numerical methods to visually illustrate the dependence of  the wavelength shift on both the initial electron momentum and the scattering angle. Furthermore, this paper discusses the underlying mechanisms and necessary conditions for the occurrence of the interference scattering and inverse Compton scattering.