Continual Machine Reading Comprehension aims to incrementally learn from a continuous data stream across time without access the previous seen data, which is crucial for the development of real-world MRC systems. However, it is a great challenge to learn a new domain incrementally without catastrophically forgetting previous knowledge. In this paper, MA-MRC, a continual MRC model with uncertainty-aware fixed Memory and Adversarial domain adaptation, is proposed. In MA-MRC, a fixed size memory stores a small number of samples in previous domain data along with an uncertainty-aware updating strategy when new domain data arrives. For incremental learning, MA-MRC not only keeps a stable understanding by learning both memory and new domain data, but also makes full use of the domain adaptation relationship between them by adversarial learning strategy. The experimental results show that MA-MRC is superior to strong baselines and has a substantial incremental learning ability without catastrophically forgetting under two different continual MRC settings.