This paper studies the problem of distribution matching (DM), which is a fundamental machine learning problem seeking to robustly align two probability distributions. Our approach is established on a relaxed formulation, called partial distribution matching (PDM), which seeks to match a fraction of the distributions instead of matching them completely. We theoretically derive the Kantorovich-Rubinstein duality for the partial Wasserstain-1 (PW) discrepancy, and develop a partial Wasserstein adversarial network (PWAN) that efficiently approximates the PW discrepancy based on this dual form. Partial matching can then be achieved by optimizing the network using gradient descent. Two practical tasks, point set registration and partial domain adaptation are investigated, where the goals are to partially match distributions in 3D space and high-dimensional feature space respectively. The experiment results confirm that the proposed PWAN effectively produces highly robust matching results, performing better or on par with the state-of-the-art methods.