In this talk, we will explore statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. This includes limit distributions for the \$L^p\$-error with arbitrary \$p \in [1,\infty)\$ and for linear functionals of the empirical OT map, together with their moment convergence, and consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. I will discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation. Connections to the entropic optimal transport problem (EOT) in the semi-discrete will also be explored.