Event extraction is a fundamental task in information extraction. Most previous approaches typically transform event extraction into two subtasks: trigger classification and argument classification, and solve them via classification-based methods, which suffer from some inherent drawbacks. To overcome these issues, in this paper we propose a novel event extraction model Seq2EG by first formulating event extraction as an event graph parsing problem, and then exploiting a pre-trained sequence-to-sequence (seq2seq) model to transduce an input sentence into an accurate event graph without the need for trigger words. Based on the generative event graph parsing formulation, our model Seq2EG can explicitly model the multiple event correlations and argument sharing, and can naturally incorporate some graph-structured features and the rich semantic information conveyed by the labels of event types and argument roles. Extensive experimental results on the public ACE2005 dataset show that, our approach outperforms all previous state-of-the-art models for event extraction by a large margin, respectively obtaining an improvement of 3.4% F1 score for event detection and an improvement of 4.7% F1 score for argument classification over the best baselines.