Neural networks are known to suffer from catastrophic forgetting when trained on sequential datasets. While there have been numerous attempts to solve this problem for large-scale supervised classification, little has been done to overcome catastrophic forgetting for few-shot classification problems. We demonstrate that the popular gradient-based few-shot meta-learning algorithm Model-Agnostic Meta-Learning (MAML) indeed suffers from catastrophic forgetting and introduce a Bayesian online meta-learning framework that tackles this problem. Our framework incorporates MAML into a Bayesian online learning algorithm with Laplace approximation. This framework enables few-shot classification on a range of sequentially arriving datasets with a single meta-learned model. The experimental evaluations demonstrate that our framework can effectively prevent forgetting in various few-shot classification settings compared to applying MAML sequentially.