Although graded rings, the grades of which are embedded in an abelian group, have been studied by Chevalley (The Construction and Study of Certain Important Algebras, 1955), the first relatively general definition of graded groups and graded rings was given by Bourbaki (1958). However, that definition was not general enough because it was based on an abelian graded group. M. Krasner started from that definition (Krasner 1980), giving up the original unnecessary restriction. Krasner’s definition, leaving aside the hypothesis of commutativity, shows that the structure of a graded group is characterized by both the underlying abstract group and its homogeneous subset (or even only by the homogeneous subset). It is well known that the classical Bourbaki–Krasner’s graded structures belong to the category that is not closed with respect to the direct product and the direct sum. The main subject of Krasner’s and my joint papers and monograph (Krasner and Vuković 1986a,b, 1987a,b) was to introduce the structures (groups, rings, modules) that generalize the corresponding graded structures and have, in each of these three cases, the property of closure with respect to the direct sum and the direct product. We called the obtained structures para- and extra-graded. The goal of this chapter is to review the basic definitions and assertions concerning graded, para-, and extra-graded structures mentioned in the title. Within the framework of Bourbaki–Krasner graduations and para- and extra-graduations, these structures were studied by M. Krasner, M. Chadeyras, E. Halberstadt, myself, and my students. All results mentioned here have been published or will be published in the papers cited in the bibliography.