Optimal Transport (OT) formulates a powerful framework by comparing probability distributions, and it has increasingly attracted great attention within the machine learning community. However, it suffers from severe computational burden, due to the intractable objective with respect to the distributions of interest. Especially, there still exist very few attempts for continuous OT, i.e., OT for comparing continuous densities. To this end, we develop a novel continuous OT method, namely Copula OT (Cop-OT). The basic idea is to transform the primal objective of continuous OT into a tractable form with respect to the copula parameter, which can be efficiently solved by stochastic optimization with less time and memory requirements. Empirical results on real applications of image retrieval and synthetic data demonstrate that our Cop-OT can gain more accurate approximations to continuous OT values than the state-of-the-art baselines.