This book deals with quantum physical calculations of anomalous scattering of photons by bound electrons. The main attention is paid to studies of the process of photon scattering in the energy range of deep levels of such many-electron systems as atoms, atomic ions and simple molecules. The book is written for scientists and graduate students, working in the field of atomic and molecular spectroscopy, physics of plasmas, lasers, solid state, ionizing radiation and astrophysics. It can also be used as additional textbook for a special topic of quantum mechanics.