We consider the problem of optimal transportation with general  cost between a empirical measure and a general target probability on $\mathbb{R}^{d}$, with $d\geq 1$. We extend results in del Barrio and Loubes and prove asymptotic stability of both optimal transport maps and potentials for a large class of costs in $\mathbb{R}^{d}$. We derive a central limit theorem (CLT) towards a Gaussian distribution for the empirical transportation cost under minimal assumptions, with a  new proof  based on the Efron-Stein inequality and on the sequential compactness  of the closed unit ball in $L_{2}(\mathbb{P})$ for the weak topology.
