We advance the state-of-the-art in the accuracy of code prediction (next token prediction) used in autocomplete systems. First, we report that using the recently proposed Transformer architecture even out-of-the-box outperforms previous neural and non-neural systems for code prediction. We then show that by making the Transformer architecture aware of the syntactic structure of code, we further increase the margin by which a Transformer-based system outperforms previous systems. With this, it outperforms the accuracy of an RNN-based system (similar to Hellendoorn et al. 2018) by 18.3%, the Deep3 system (Raychev et al 2016) by 14.1%, and an adaptation of Code2Seq (Alon et al., 2018) for code prediction by 14.4%. We present in the paper several ways of communicating the code structure to the Transformer, which is fundamentally built for processing sequence data. We provide a comprehensive experimental evaluation of our proposal, along with alternative design choices, on a standard Python dataset, as well as on a Facebook internal Python corpus. Our code and data preparation pipeline will be available in open source.