Continual Learning (CL) has achieved rapid progress in recent years. However, it is still largely unknown how to determine whether a CL model is trustworthy and how to foster its trustworthiness. This work focuses on evaluating and improving the robustness to corruptions of existing CL models. Our empirical evaluation results show that existing state-of-the-art (SOTA) CL models are particularly vulnerable to various data corruptions during testing. To make them trustworthy and robust to corruptions deployed in safety-critical scenarios, we propose a meta-learning framework of self-adaptive data augmentation to tackle the corruption robustness in CL. The proposed framework, MetaMix, learns to augment and mix data, automatically transforming the new task data or memory data. It directly optimizes the generalization performance against data corruptions during training. To evaluate the corruption robustness of our proposed approach, we construct several CL corruption datasets with different levels of severity. We perform comprehensive experiments on both task- and class-continual learning. Extensive experiments demonstrate the effectiveness of our proposed method compared to SOTA baselines.