  The aim of this article is to review different generalizations of the the
notion of topological complexity to the equivariant setting. In particular, we
review the relation (or non-relation) between these notions and the topological
complexity of the quotient space and the topological complexity of the fixed
point sets. We give examples of calculations and stress the question: When the
action is free, do we recover the topological complexity of the quotient?
