Abstract: We present recent approaches to prove the asymptotic behaviour of empirical transport cost, $\mathcal{T}_c(P_n,Q)$, under minimal assumptions in high dimension. Centering around its expectation, the weak limit of $\sqrt{n}\{\mathcal{T}_c(P_n,Q)-E\mathcal{T}_c(P_n,Q)\}$ is Gaussian. Yet, due to the curse of dimensionality, the variable $E\mathcal{T}_c(P_n,Q)$ can not be exchanged by its population counterpart $\mathcal{T}_c(P,Q)$. When $P$ is finitely supported this problem can be solved and the limit becomes the supremum of a centered Gaussian process, which is Gaussian under some additional conditions on the probability $Q$.