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Traffic forecasting can be highly challenging due to complex spatial-temporal correlations and non-linear traffic patterns. Existing works mostly model such spatial-temporal dependencies by considering spatial correlations and temporal correlations separately, or within a sliding temporal window, and fail to model the direct spatial-temporal correlations. Inspired by the recent success of transformers in the graph domain, in this paper, we propose to directly model the cross-spatial-temporal correlations on the adaptive spatial-temporal graph using local multi-head self-attentions. We then propose a novel Adaptive Graph Spatial-Temporal Transformer Network (ASTTN), which stacks multiple spatial-temporal attention layers to apply self-attention on the input graph, followed by linear layers for predictions. Experimental results on public traffic network datasets, METR-LA PEMS-BAY, PeMSD4, and PeMSD7 …