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Although a number of studies are devoted to novel category discovery, most of them assume a static setting where both labeled and unlabeled data are given at once for finding new categories. In this work, we focus on the application scenarios where unlabeled data are continuously fed into the category discovery system. We refer to it as the {\bf Continuous Category Discovery} ({\bf CCD}) problem, which is significantly more challenging than the static setting. A common challenge faced by novel category discovery is that different sets of features are needed for classification and category discovery: class discriminative features are preferred for classification, while rich and diverse features are more suitable for new category mining. This challenge becomes more severe for dynamic setting as the system is asked to deliver good performance for known classes over time, and at the same time continuously discover new classes from unlabeled data. To address this challenge, we develop a framework of {\bf Grow and Merge} ({\bf GM}) that works by alternating between a growing phase and a merging phase: in the growing phase, it increases the diversity of features through a continuous self-supervised learning for effective category mining, and in the merging phase, it merges the grown model with a static one to ensure satisfying performance for known classes. Our extensive studies verify that the proposed GM framework is significantly more effective than the state-of-the-art approaches for continuous category discovery.