Adversarial training algorithms have been proved to be reliable to improve machine learning models' robustness against adversarial examples. However, we find that adversarial training algorithms tend to introduce severe disparity of accuracy and robustness between different groups of data. For instance, a PGD adversarially trained ResNet18 model on CIFAR-10 has 93% clean accuracy and 67% PGD l-infty-8 robust accuracy on the class "automobile" but only 65% and 17% on the class "cat". This phenomenon happens in balanced datasets and does not exist in naturally trained models when only using clean samples. In this work, we empirically and theoretically show that this phenomenon can happen under general adversarial training algorithms which minimize DNN models' robust errors. Motivated by these findings, we propose a Fair-Robust-Learning (FRL) framework to mitigate this unfairness problem when doing adversarial defenses. Experimental results validate the effectiveness of FRL.