It is easy to check that the sum of any family of two-sided nil ideals of an associative ring is a nil ideal as well. Does the same hold for left nil ideals? Though this question looks very elementary and was raised long ago (in 1930 by Koethe) it is still open. It is called Koethe’s nil ideal problem and is one of the most famous open problems in ring theory. Attempts to solve this problem led to many interesting, deep and sometimes surprising results. There are also many related open problems. The aim of the talk is to present some new results obtained in the area.