Speech distortions are a long-standing problem that degrades the performance of supervisely trained speech processing models. It is high time that we enhance the robustness of speech processing models to obtain good performance when encountering speech distortions while not hurting the original performance on clean speech. In this work, we propose to improve the robustness of speech processing models by domain adversarial training (DAT). We conducted experiments based on the SUPERB framework on five different speech processing tasks. In case we do not always have knowledge of the distortion types for speech data, we analyzed the binary-domain and multi-domain settings, where the former treats all distorted speech as one domain, and the latter views different distortions as different domains. In contrast to supervised training methods, we obtained promising results in target domains where speech data is distorted with different distortions including new unseen distortions introduced during testing.