A reliable autonomous driving system should take safe and efficient actions in constantly changing traffic. This requires the trajectory prediction model to continuously learn from incoming data and adapt to new scenarios. In the context of rapidly growing data volume, existing trajectory prediction models must retrain on all datasets to avoid forgetting previously learned knowledge when facing additional data from new environments. In contrast, the paradigm of continual learning solely necessitates training on new data, saving a significant amount of training overhead. Therefore, it is crucial to equip the trajectory prediction model with the ability of continual learning. In this paper, inspired by rehearsal and pseudo-rehearsal methods in continual learning, we propose a continual trajectory prediction framework with uncertainty-aware generative memory replay, CTP-UGR. Our framework effectively avoids excessive memory space requirements while generating trajectory data that is authentic, representative and discriminative for continual learning. Extensive experiments on two real-world datasets demonstrate our proposed CTP-UGR significantly outperforms other baselines in terms of both accuracy and catastrophic forgetting. Besides, our framework can be combined with other state-of-the-art trajectory prediction models to achieve better performance.