  Hyperelliptic mapping class groups are defined either as the centralizers of
hyperelliptic involutions inside mapping class groups of oriented surfaces of
finite type or as the inverse images of these centralizers by the natural
epimorphisms between mapping class groups of surfaces with marked points. We
study these groups in a systematic way. An application of this theory is a
counterexample to the genus $2$ case of a conjecture by Putman and Wieland on
virtual linear representations of mapping class groups. In the last section, we
study profinite completions of hyperelliptic mapping class groups: we extend
the congruence subgroup property to the general class of hyperelliptic mapping
class groups introduced above and then determine the centralizers of
multitwists and of open subgroups in their profinite completions.
