This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in the volumes 121 (Part I, Chaps. 1–4, Secs. 1–22) and 122 (Part II, Chaps. 5–8, Secs. 23–39) of the series Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory (English translation: Journal of Mathematical Sciences (New York), vol. 145, No. 4, pp. 4997–5117 (2007), and vol. 151, No. 5, pp. 3225–3371 (2008)). In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 “Appendix,” Secs. 40–42. In Sec. 40, we consider p-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings. The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results