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Generally, multimodal data with new classes arrive continuously in the real world. While advanced cross-modal hashing (CMH) focuses primarily on batch-based data with previously observed classes (ASCs), it disregards the effect of newly arriving classes (ANCs) on hash-code conflicts. In addition, class-level continuous hashing scenarios do not suit themselves well with the generic CMH configuration. To solve the aforementioned issues, we propose a novel framework, called CT-CMH, for the new task of continuous cross-modal hashing. For dealing with ANCs, CMH models require the ability of continuous learning, i.e. they can preserve the knowledge of previously observed data and, more crucially, they can be adapted to unseen data with ANCs. Specifically, we introduce the adaptive weight importance updating (AWIU) mechanism to alleviate the catastrophic forgetting problem of CMH and a new hash-code …