Lorentz symmetry is one of the fundamental symmetries in modern physics, however, certain models of quantum gravity point to the possibility of its violation. One possible way of detecting Lorentz symmetry breaking is provided by photon–photon scattering of gamma-rays on the extragalactic background radiation. In this thesis, photon–photon scattering is analyzed in the context of the standard and extended theory of quantum electrodynamics. An overview of the derivation of Fermi’s golden rule and the cross section of photon–photon scattering in standard quantum electrodynamics is given. The analysis of the cross section with the modified reaction threshold is done. Using the model of extended quantum electrodynamics, the influence of dynamic changes on the impact cross section for photon–photon scattering is studied. The cross section in the ultra–relativistic approximation is analyzed. The standard cross section in the laboratory system is calculated and the general form of the cross section of two-particle scattering is given. Applying the obtained results, the ultra–relativistic approximation is discarded, which makes the calculation for the modified cross section for photon—photon scattering more accurate in the range of all energies. The kinematics of scattering in a laboratory system was studied.