This manuscript aims to systematically consider the main periodicity and additional (secondary, internal, and tetrad) periodicities using a uniform approach. The main features are summarized in table form. The history of the origin and development of these concepts is discussed. It is described how these periodicities manifest themselves and how they are determined at the experimental and theoretical levels. Areas of manifestation of these periodicities are outlined. As the general approach to explaining internal periodicity, attention is drawn to the symmetry (...) of the quantum number S of atoms and the principle of equivalence of electrons and holes. Arguments are presented in favor of a more correct classification of the tetrad effect as tetrad periodicity, and an overview of this regularity is provided. A small modification of the conventional Periodic table is proposed, which reflects all the mentioned periodicities. (shrink)