Optimal Transport (OT) is a resource allocation problem with applications in biology, data science, economics and statistics, among others. In some of the applications, practitioners have access to samples which approximate the continuous measure. Hence the quantities of interest derived from OT -- plans, maps and costs -- are only available in their empirical versions. Statistical inference on OT aims at finding confidence intervals of the population plans, maps and costs. In recent years this topic gained an increasing interest in the statistical community. In this paper we provide a comprehensive review of the most influential results on this research field, underlying the some of the applications. Finally, we provide a list of open problems.
Barrio et al. (Sun,) studied this question.