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 more than seventy years ago (in 1930 by Köthe) it is still open. It is called Köthe's nil ideal problem and is one of the most famous open problems in ring theory. Attempts to solve it led to many interesting, deep and sometimes surprising results. There are also many related open problems. The aim of the talk is to present several such problems as well as some old and new results obtained in the area. In particular it will be shown that Köthe's problem is equivalent to a problem raised in 1969 by Andrunakievich.