There has been a surge of interest in continual learning and federated learning, both of which are important in training deep neural networks in real-world scenarios. Yet little research has been done regarding the scenario where each client learns on a sequence of tasks from private local data. This problem of federated continual learning poses new challenges to continual learning, such as utilizing knowledge and preventing interference from tasks learned on other clients. To resolve these issues, we propose a novel federated continual learning framework, Federated continual learning with Adaptive Parameter Communication, which additively decomposes the network weights into global shared parameters and sparse task-specific parameters. This decomposition allows to minimize interference between incompatible tasks, and also allows inter-client knowledge transfer by communicating the sparse task-specific parameters. Our federated continual learning framework is also communication-efficient, due to high sparsity of the parameters and sparse parameter update. We validate APC against existing federated learning and local continual learning methods under varying degrees of task similarity across clients, and show that our model significantly outperforms them with a large reduction in the communication cost.