Adversarial training is one of the most effective approaches for deep learning models to defend against adversarial examples. Unlike other defense strategies, adversarial training aims to enhance the robustness of models intrinsically. During the past few years, adversarial training has been studied and discussed from various aspects, which deserves a comprehensive review. For the first time in this survey, we systematically review the recent progress on adversarial training for adversarial robustness with a novel taxonomy. Then we discuss the generalization problems in adversarial training from three perspectives and highlight the challenges which are not fully tackled. Finally, we present potential future directions.