We propose a novel method, namely the accelerated mirror-prox (AMP) method, for computing the weak solutions of a class of deterministic and stochastic monotone variational inequalities (VI). The main idea of this algorithm is to incorporate a multi-step acceleration scheme into the mirror-prox method. For both deterministic and stochastic VIs, the developed AMP method computes the weak solutions with optimal rate of convergence. In particular, if the monotone operator in VI consists of the gradient of a smooth function, the rate of convergence of the AMP method can be accelerated in terms of its dependence on the Lipschitz constant of the smooth function. For VIs with bounded feasible sets, the estimate of the rate of convergence of the AMP method depends on the diameter of the feasible set. For unbounded VIs, we adopt the modified gap function introduced by Monteiro and Svaiter for solving monotone inclusion, and demonstrate that the rate of convergence of the AMP method depends on the distance from the initial point to the set of strong solutions.