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Query-aware webpage snippet extraction is widely used in search engines to help users better understand the content of the returned webpages before clicking. Although important, it is very rarely studied. In this paper, we propose an effective query-aware webpage snippet extraction method named DeepQSE, aiming to select a few sentences which can best summarize the webpage content in the context of input query. DeepQSE first learns query-aware sentence representations for each sentence to capture the fine-grained relevance between query and sentence, and then learns document-aware query-sentence relevance representations for snippet extraction. Since the query and each sentence are jointly modeled in DeepQSE, its online inference may be slow. Thus, we further propose an efficient version of DeepQSE, named Efficient-DeepQSE, which can significantly improve the inference speed of DeepQSE without affecting its performance. The core idea of Efficient-DeepQSE is to decompose the query-aware snippet extraction task into two stages, i.e., a coarse-grained candidate sentence selection stage where sentence representations can be cached, and a fine-grained relevance modeling stage. Experiments on two real-world datasets validate the effectiveness and efficiency of our methods.