In the history of quantum theory the Compton effect has played an important role providing one of the direct proofs of the corpuscular nature of the electromagnetic radiation. The book is devoted to a theoretical study of laser assisted Compton scattering on the ground state of a hydrogenlike atom. Based on analytic formula, valid for any fixed polarization of the laser, and derived by the author using the nonrelativistic Coulomb-Volkov function for the final electron, one presents as graphs the results of numerical calculations for the electron energy spectrum at a variety of fixed photon-electron configurations and for the distribution of electrons in the case in which the scattered photon is not detected. Useful appendices accompany the main text. Results could be useful in the context in which the Compton effect is the basic process used in designing sources of intense high frequency electromagnetic radiation, dedicated to many applications in science, technology and medicine. The fascinating progress in laser and electron detection technology will continue to stimulate detailed theoretical studies of photon-bound electron interaction.