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In this paper, we propose a general framework in continual learning for generative models: Feature-oriented Continual Learning (FoCL). Unlike previous works that aim to solve the catastrophic forgetting problem by introducing regularization in the parameter space or image space, FoCL imposes regularization in the feature space. We show in our experiments that FoCL has faster adaptation to distributional changes in sequentially arriving tasks, and achieves the state-of-the-art performance for generative models in task incremental learning. We discuss choices of combined regularization spaces towards different use case scenarios for boosted performance, e.g., tasks that have high variability in the background. Finally, we introduce a forgetfulness measure that fairly evaluates the degree to which a model suffers from forgetting. Interestingly, the analysis of our proposed forgetfulness score also implies that FoCL tends to have a mitigated forgetting for future tasks.