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Grasping moving objects, such as goods on a belt or living animals, is an important but challenging task in robotics. Conventional approaches rely on a set of manually defined object motion patterns for training, resulting in poor generalization to unseen object trajectories. In this work, we introduce an adversarial reinforcement learning framework for dynamic grasping, namely GraspARL. To be specific. we formulate the dynamic grasping problem as a 'move-and-grasp' game, where the robot is to pick up the object on the mover and the adversarial mover is to find a path to escape it. Hence, the two agents play a min-max game and are trained by reinforcement learning. In this way, the mover can auto-generate diverse moving trajectories while training. And the robot trained with the adversarial trajectories can generalize to various motion patterns. Empirical results on the simulator and real-world scenario demonstrate the effectiveness of each and good generalization of our method.