  For an algebraic stack $\sX$ flat and of finite presentation over a scheme
$S$, we introduce various notions of {\em relative connected components} and
{\em relative irreducible components}. The main distinction between these
notions is whether we require the total space of a relative component to be
open or closed in $\sX$. We study the representability of the associated
functors of relative components, and give an application to the moduli stack of
curves of genus $g$ admitting an action of a fixed finite group $G$.
