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Deep Neural Networks (DNNs) have been ubiquitously adopted in internet of things and are becoming an integral of our daily life. When tackling the evolving learning tasks in real world, such as classifying different types of objects, DNNs face the challenge to continually retrain themselves according to the tasks on different edge devices. Federated continual learning is a promising technique that offers partial solutions but yet to overcome the following difficulties: the significant accuracy loss due to the limited on-device processing, the negative knowledge transfer caused by the limited communication of non-IID data, and the limited scalability on the tasks and edge devices. In this paper, we propose FedKNOW, an accurate and scalable federated continual learning framework, via a novel concept of signature task knowledge. FedKNOW is a client side solution that continuously extracts and integrates the knowledge of signature tasks which are highly influenced by the current task. Each client of FedKNOW is composed of a knowledge extractor, a gradient restorer and, most importantly, a gradient integrator. Upon training for a new task, the gradient integrator ensures the prevention of catastrophic forgetting and mitigation of negative knowledge transfer by effectively combining signature tasks identified from the past local tasks and other clients' current tasks through the global model. We implement FedKNOW in PyTorch and extensively evaluate it against state-of-the-art techniques using popular federated continual learning benchmarks. Extensive evaluation results on heterogeneous edge devices show that FedKNOW improves model accuracy by 63.24% without increasing model training time, reduces communication cost by 34.28%, and achieves more improvements under difficult scenarios such as large numbers of tasks or clients, and training different complex networks.