  In this paper, an approach for generalizing the Gromov-Hausdorff metric is
presented, which applies to metric spaces equipped with some additional
structure. A special case is the Gromov-Hausdorff-Prokhorov metric between
measured metric spaces. This abstract framework unifies several existing
Gromov-Hausdorff-type metrics for metric spaces equipped with a measure, a
point, a closed subset, a curve, a tuple of such structures, etc. Along with
reviewing these special cases in the literature, several new examples are also
presented. Two frameworks are provided, one for compact metric spaces and the
other for boundedly-compact pointed metric spaces. In both cases, a
Gromov-Hausdorff-type metric is defined and its topological properties are
studied. In particular, completeness and separability is proved under some
conditions. This enables one to study random metric spaces equipped with
additional structures, which is the main motivation of this work.
