Machine reading comprehension (MRC) has become a core component in a variety of natural language processing (NLP) applications such as question answering and dialogue systems. It becomes a practical challenge that an MRC model needs to learn in non-stationary environments, in which the underlying data distribution changes over time. A typical scenario is the domain drift, i.e. different domains of data come one after another, where the MRC model is required to adapt to the new domain while maintaining previously learned ability. To tackle such a challenge, in this work, we introduce the \textit{Continual Domain Adaptation} (CDA) task for MRC. So far as we know, this is the first study on the continual learning perspective of MRC. We build two benchmark datasets for the CDA task, by re-organizing existing MRC collections into different domains with respect to context type and question type, respectively. We then analyze and observe the catastrophic forgetting (CF) phenomenon of MRC under the CDA setting. To tackle the CDA task, we propose several BERT-based continual learning MRC models using either regularization-based methodology or dynamic-architecture paradigm. We analyze the performance of different continual learning MRC models under the CDA task and show that the proposed dynamic-architecture based model achieves the best performance.