Recent years have witnessed the rapid development of online social platforms, which effectively support the business intelligence and provide services for massive users. Along this line, large efforts have been made on the socialaware recommendation task, i.e., leveraging social contextual information to improve recommendation performance. Most existing methods have treated social relations in a static way, but the dynamic influence of social contextual information on users' consumption choices has been largely unexploited. To that end, in this paper, we conduct a comprehensive study to reveal the dynamic social influence on users' preferences, and then we propose a deep model called Dynamic Social-Aware Recommender System (DSRS) to integrate the users' structural and temporal social contexts to address the dynamic socialaware recommendation task. DSRS consists of two main components, i.e., the social influence learning (SIL) and dynamic preference learning (DPL). Specifically, in the SIL module, we arrange social graphs in a sequential order and borrow the power of graph convolution networks (GCNs) to learn social context. Moreover, we design a structural-temporal attention mechanism to discriminatively model the structural social influence and the temporal social influence. Then, in the DPL part, users' individual preferences are learned dynamically by recurrent neural networks (RNNs). Finally, with a prediction layer, we combine the users' social context and dynamic preferences to generate recommendations. We conduct extensive experiments on two real-world datasets, and the experimental results demonstrate the superiority and effectiveness of our proposed model compared with the state-of-the-art methods.