  Let $w$ be a group-word. Suppose that the set of all $w$-values in a
profinite group $G$ is contained in a union of countably many subgroups. It is
natural to ask in what way the structure of the verbal subgroup $w(G)$ depends
on the properties of the covering subgroups. The present article is a survey of
recent results related to that question. In particular we survey results on
finite and countable coverings of word-values (mostly commutators) by
procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with
finiteness conditions. The last section of the paper is devoted to relation of
the described results with Hall's problem on conciseness of group-words.
