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A right Engel sink of an element g of a group G is a set 
${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators 
$[...[[g,x],x],\dots ,x]$ belong to 
${\mathscr R}(g)$. (Thus, g is a right Engel element precisely when we can choose 
${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.
Let G be a linear group such that for every g ∈ G there is a finite set 
${\cal R}(g)$ with the property that for every x ∈ G all sufficiently long commutators [g, x, x, …, x] belong to 
${\cal R}(g)$. We prove that G is finite-by-hypercentral.
