  In 1995 in Kourovka notebook the second author asked the following problem:
it is true that for each partition $G=A_1\cup\dots\cup A_n$ of a group $G$
there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some set
$F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial
solutions of this problem, in particular those involving certain canonical
invariant densities and submeasures on groups.
