  Homotopical localizations with respect to a set of maps are known to exist in
cofibrantly generated model categories (satisfying additional assumptions). In
this paper we expand the existing framework, so that it will apply to not
necessarily cofibrantly generated model categories and, more important, will
allow for a localization with respect to a class of maps (satisfying some
restrictive conditions).
  We illustrate our technique by applying it to the equivariant model category
of diagrams of spaces. This model category is not cofibrantly generated. We
give conditions on a class of maps which ensure the existence of the
localization functor; these conditions are satisfied by any set of maps and by
the class of maps which induces ordinary localizations on the generalized
fixed-points sets.
