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Taxi demand forecasting plays an important role in ride-hailing services. Accurate taxi demand forecasting can assist taxi companies in pre-allocating taxis, improving vehicle utilization, reducing waiting time, and alleviating traffic congestion. It is a challenging task due to the highly non-linear and complicated spatial-temporal patterns of the taxi data. Most of the existing taxi demand forecasting methods lack the ability to capture the dynamic spatial-temporal dependencies among regions. They either fail to consider the limitations of Graph Neural Networks or do not efficiently capture the long-term temporal dependencies. In this paper, we propose a Spatial-Temporal Diffusion Convolutional Network (ST-DCN) for taxi demand forecasting. The dynamic spatial dependencies are efficiently captured through a two-phase graph diffusion convolutional network where the attention mechanism is introduced. Moreover, a novel temporal convolution module is designed to learn various ranges of temporal dependencies, including recent, daily, and weekly periods. Inside the module, convolution layers are stacked to handle very long sequences. Experimental results on two large-scale real-world taxi datasets from New York City (NYC) and Chengdu demonstrate that our method significantly outperforms seven state-of-the-art baseline methods. [ABSTRACT FROM AUTHOR]