In this talk, we propose a new approximation method to solve distributionally robust optimization problems with moment-based ambiguity sets. Our approximation method relies on principal component analysis (PCA) for optimal lower dimensional representation of variability in random samples. We show that the PCA approximation yields a relaxation of the original problem and derive theoretical bounds on the gap between the original problem and its PCA approximation. Furthermore, an extensive numerical study shows the strength of the proposed approximation method in terms of solution quality and runtime.