Given two point sets, the problem of registration is to recover a transformation that matches one set to the other. This task is challenging due to the presence of large number of outliers, the unknown non-rigid deformations and the large sizes of point sets. To obtain strong robustness against outliers, we formulate the registration problem as a partial distribution matching (PDM) problem, where the goal is to partially match the distributions represented by point sets in a metric space. To handle large point sets, we propose a scalable PDM algorithm by utilizing the efficient partial Wasserstein-1 (PW) discrepancy. Specifically, we derive the Kantorovich-Rubinstein duality for the PW discrepancy, and show its gradient can be explicitly computed. Based on these results, we propose a partial Wasserstein adversarial network (PWAN), which is able to approximate the PW discrepancy by a neural network, and minimize it by gradient descent. In addition,it also incorporates an efficient coherence regularizer for non-rigid transformations to avoid unrealistic deformations. We evaluate PWAN on practical point set registration tasks, and show that the proposed PWAN is robust, scalable and performs more favorably than the state-of-the-art methods.