Flow prediction (e.g., crowd flow, traffic flow) with features of spatial-temporal is increasingly investigated in AI research field. It is very challenging due to the complicated spatial dependencies between different locations and dynamic temporal dependencies among different time intervals. Although measurements of both dependencies are employed, existing methods suffer from the following two problems. First, the temporal dependencies are measured either uniformly or bias against long-term dependencies, which overlooks the distinctive impacts of short-term and long-term temporal dependencies. Second, the existing methods capture spatial and temporal dependencies independently, which wrongly assumes that the correlations between these dependencies are weak and ignores the complicated mutual influences between them. To address these issues, we propose a Spatial-Temporal Self-Attention Network (ST-SAN). As the path-length of attending long-term dependency is shorter in the self-attention mechanism, the vanishing of long-term temporal dependencies is prevented. In addition, since our model relies solely on attention mechanisms, the spatial and temporal dependencies can be simultaneously measured. Experimental results on real-world data demonstrate that, in comparison with state-of-the-art methods, our model reduces the root mean square errors by 9% in inflow prediction and 4% in outflow prediction on Taxi-NYC data, which is very significant compared to the previous improvement.