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Continual learning with neural networks is an important learning framework in AI that aims to learn a sequence of tasks well. However, it is often confronted with three challenges: (1) overcome the catastrophic forgetting problem, (2) adapt the current network to new tasks, and meanwhile (3) control its model complexity. To reach these goals, we propose a novel approach named as Continual Learning with Efficient Architecture Search, or CLEAS in short. CLEAS works closely with neural architecture search (NAS) which leverages reinforcement learning techniques to search for the best neural architecture that fits a new task. In particular, we design a neuron-level NAS controller that decides which old neurons from previous tasks should be reused (knowledge transfer), and which new neurons should be added (to learn new knowledge). Such a fine-grained controller allows one to find a very concise architecture that can fit each new task well. Meanwhile, since we do not alter the weights of the reused neurons, we perfectly memorize the knowledge learned from previous tasks. We evaluate CLEAS on numerous sequential classification tasks, and the results demonstrate that CLEAS outperforms other state-of-the-art alternative methods, achieving higher classification accuracy while using simpler neural architectures.