This article is a continuation of a series of surveys on distributive and semidistributive rings and modules [\textit{A. V. Mikhalev, A. A. Tuganbaev}, J. Math. Sci., New York 93, No.~2, 149-253 (1999; Zbl 0928.16003); ibid. 94, No.~6, 1809-1887, 1888-1924 (1999; Zbl 0936.16007)]. This time in the focus of attention are flat modules, distributive rings, and their relations. Some characterisations of flat modules are mentioned, then conditions are shown when a factor module of a flat module is flat, and when the class of flat modules is hereditary. Descriptions of pf-rings (principal left ideals are flat) and of distributive modules over right invariant rings are incluced in this survey. The principal classes of rings studied in this work are: distributive rings, coherent rings, invariant rings, Bézout rings, algebraic (module finite) over their center. The greater part of the present article was included in the author's book ``Semidistributive modules and rings'' [Kluwer Acad. Publ., Math. Appl. 449 (1998; Zbl 0909.16001)].