  Let $w$ be a multilinear commutator word. In the present paper we describe
recent results that show that if $G$ is a profinite group in which all
$w$-values are contained in a union of finitely (or in some cases countably)
many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has
the same property as well. In particular, we show this in the case where the
subgroups are periodic or of finite rank.
