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Adversarial training is a defense technique that improves adversarial robustness of a deep neural network (DNN) by including adversarial examples in the training data. In this paper, we identify an overlooked problem of adversarial training in that these adversarial examples often have different semantics than the original data, introducing unintended biases into the model. We hypothesize that such non-semantics-preserving (and resultingly ambiguous) adversarial data harm the robustness of the target models. To mitigate such unintended semantic changes of adversarial examples, we propose semantics-preserving adversarial training (SPAT) which encourages perturbation on the pixels that are shared among all classes when generating adversarial examples in the training stage. Experiment results show that SPAT improves adversarial robustness and achieves state-of-the-art results in CIFAR-10 and CIFAR-100.