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Class-incremental learning (CIL) suffers from the notorious dilemma between learning newly added classes and preserving previously learned class knowledge. That catastrophic forgetting issue could be mitigated by storing historical data for replay, which yet would cause memory overheads as well as imbalanced prediction updates. To address this dilemma, we propose to leverage "free" external unlabeled data querying in continual learning. We first present a CIL with Queried Unlabeled Data (CIL-QUD) scheme, where we only store a handful of past training samples as anchors and use them to query relevant unlabeled examples each time. Along with new and past stored data, the queried unlabeled are effectively utilized, through learning-without-forgetting (LwF) regularizers and class-balance training. Besides preserving model generalization over past and current tasks, we next study the problem of adversarial robustness for CIL-QUD. Inspired by the recent success of learning robust models with unlabeled data, we explore a new robustness-aware CIL setting, where the learned adversarial robustness has to resist forgetting and be transferred as new tasks come in continually. While existing options easily fail, we show queried unlabeled data can continue to benefit, and seamlessly extend CIL-QUD into its robustified versions, RCIL-QUD. Extensive experiments demonstrate that CIL-QUD achieves substantial accuracy gains on CIFAR-10 and CIFAR-100, compared to previous state-of-the-art CIL approaches. Moreover, RCIL-QUD establishes the first strong milestone for robustness-aware CIL. Codes are available in https://github.com/VITA-Group/CIL-QUD.