Event extraction, which aims to identify event triggers of pre-defined event types and their arguments of specific roles, is a challenging task in NLP. Most traditional approaches formulate this task as classification problems, with event types or argument roles taken as golden labels. Such approaches fail to model rich interactions among event types and arguments of different roles, and cannot generalize to new types or roles. This work proposes a new paradigm that formulates event extraction as multi-turn question answering. Our approach, MQAEE, casts the extraction task into a series of reading comprehension problems, by which it extracts triggers and arguments successively from a given sentence. A history answer embedding strategy is further adopted to model question answering history in the multi-turn process. By this new formulation, MQAEE makes full use of dependency among arguments and event types, and generalizes well to new types with new argument roles. Empirical results on ACE 2005 shows that MQAEE outperforms current state-of-the-art, pushing the final F1 of argument extraction to 53.4% (+2.0%). And it also has a good generalization ability, achieving competitive performance on 13 new event types even if trained only with a few samples of them.