Title: Robust Weight Perturbation for Adversarial Training. Abstract: Overfitting widely exists in adversarial robust training of deep networks. An effective and promising remedy is adversarial weight perturbation, which injects the worst-case weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness enhancement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that deep network first overfits the adversarial examples with small loss, and then gradually develops to overfit all adversarial examples in the later stage of training. Following this, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate overfitting in adversarial training. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation,  significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods.