This paper studies continual learning (CL) of a sequence of aspect sentiment classification (ASC) tasks. Although some CL techniques have been proposed for document sentiment classification, we are not aware of any CL work on ASC. A CL system that incrementally learns a sequence of ASC tasks should address the following two issues: (1) transfer knowledge learned from previous tasks to the new task to help it learn a better model, and (2) maintain the performance of the models for previous tasks so that they are not forgotten. This paper proposes a novel capsule network based model called B-CL to address these issues. B-CL markedly improves the ASC performance on both the new task and the old tasks via forward and backward knowledge transfer. The effectiveness of B-CL is demonstrated through extensive experiments.