Abstract: We propose the Graph2Graph Transformer architecture for conditioning on and predicting arbitrary graphs, and apply it to the challenging task of transition-based dependency parsing. After proposing two novel Transformer models of transition-based dependency parsing as strong baselines, we show that adding the proposed mechanisms for conditioning on and predicting graphs of Graph2Graph Transformer results in significant improvements, both with and without BERT pre-training. The novel baselines and their integration with Graph2Graph Transformer significantly outperform the state-of-the-art in traditional transition-based dependency parsing on both English Penn Treebank, and 13 languages of Universal Dependencies Treebanks. Graph2Graph Transformer can be integrated with many previous structured prediction methods, making it easy to apply to a wide range of NLP tasks.