Gromov defined various notions of convergence for a sequence of compact or non-compact metric spaces, that formalize the intuition that the scaled lattice $\delta \mathbb{Z}^d$ converges to $\mathbb{R}^d$ as $\delta\to 0$. Such convergence has applications in group theory and geometry. It has also found important applications in probability theory. Most notably, Gromov-Hausdorff convergence is used in the study of limits of random graphs and other types of discrete structures, which is still a very active topic. In various applications, the metric spaces under study are equipped with additional structures (e.g., a measure) and it is required to study their limit as well. Such an additional structure can be a measure, a point or finitely many points, marks on points, a closed subset, a continuous curve, a cadlag curve, etc. Various recent papers considered such additional structures and, to study their limit, generalized the Gromov-Hausdorff metric (or convergence) appropriately. The generalizations use the same idea as Gromov’s, but are not implied by that. Therefore, the mentioned papers either stated and proved all results (being a metric, completeness, separability, and convergence criteria) or mentioned them without proof. The former required a lot of space and was not the main goal of the papers, and the latter had errors in some papers due to various technicalities. In this presentation, after a review of the literature, we present a unified framework for generalizing the Gromov-Hausdorff metric for future use by using the notion of functors. This work is based on the preprint arxiv.org/abs/1812.03760.