The monograph gives and studies axiomatic definitions of the concepts of purity, torsion and completeness (divisibility), which play an important role in the theory of Abelian groups. Recently, various generalizations of these concepts to modules have appeared in the literature.. Almost all of these generalizations fit into the scheme proposed in the monograph .. The purpose of the monograph is to summarize the progress in this area and create for further research. The presentation widely uses the methods of homologous algebra. The monograph is of interest to researchers, graduate students and students specializing in the field of algebra.