Gromov has defined a metric on the set of all compact metric spaces. This metric is defined for group-theoretic purposes but has found important applications in probability theory as well. Also, there exist several generalizations of this metric. For instance, the Gromov-Hausdorff-Prokhorov metric defines the distance of two measured metric spaces. Other examples consider metric spaces equipped with a distinguished point, a closed subset, a curve, a tuple of such structures, etc. In this talk, a general approach is presented for generalizing the Gromov-Hausdorff metric to consider metric spaces equipped with some additional structure. This abstract framework unifies the existing generalizations in the literature. It is also useful for studying new examples of additional structures which will be needed in the future works of the author. The framework is provided both for compact metric spaces and for boundedly-compact pointed metric spaces. In addition, completeness and separability of the metric are proved under some conditions. This enables us to study random metric spaces equipped with additional structures, which is the main motivation of this work.