Discrete approximation is the standard approach in stochastic programming and has been extended to distributionally robust optimization (DRO) in the past decade. In this talk, we present some quantitative stability analysis of discrete approximation schemes for DRO, which may be used to balance the approximate error and the sample size. We start by quantifying the discrepancy between the discretized ambiguity set and the original one under the Wasserstein metric when the ambiguity set is defined through equality and inequality moment conditions. The new quantitative convergence improves the existing results in two aspects: releasing the Slater type conditions and quantifying the convergence under Wasserstein metric rather than Total variation metric. Those technical results lay a theoretical foundation for various discrete approximation schemes on DRO problems.