Among C R Rao’s many contributions to statistical inference, one which has been and still is considered to be of extreme importance in the areas of statistics, physics and of signal processing in electrical engineering beside other sciences is an inequality which is now known as the Cramér–Rao inequality. This result was studied in recent years by several other scientists to relax the conditions under which it holds and to generalize it in different directions. Contributions by Bhattacharya (Sankhya 8 (1946) 1–14, 201–218, 315–328), Barankin (Ann. Math. Statist. 20 (1949) 477–501) and Fabian and Hannan (Ann. Statist. 5 (1977) 197–205) are significant in this area. We do not propose to give an extensive survey of results connected with the inequality. Our aim in this communication is to highlight some recent advances.