Polyp segmentation from colonoscopy videos is of great importance for improving the quantitative analysis of colon cancer. However, it remains a challenging task due to (1) the large size and shape variation of polyps, (2) the low contrast between polyps and background, and (3) the inherent real-time requirement of this application, where the segmentation results should be immediately presented to the doctors during the colonoscopy procedures for their prompt decision and action. It is difficult to develop a model with powerful representation capability, yielding satisfactory segmentation results in a real-time manner. We propose a novel and efficient context-aware network, named PolypSeg, in order to comprehensively address these challenges. The proposed PolypSeg consists of two key components: adaptive scale context module (ASCM) and semantic global context module (SGCM). The ASCM aggregates the multi-scale context information and takes advantage of an improved attention mechanism to make the network focus on the target regions and hence improve the feature representation. The SGCM enriches the semantic information and excludes the background noise in the low-level features, which enhances the feature fusion between high-level and low-level features. In addition, we introduce the deep separable convolution into our PolypSeg to replace the traditional convolution operations in order to reduce parameters and computational costs to make the PolypSeg run in a real-time manner. We conducted extensive experiments on a famous public available dataset for polyp segmentation task. Experimental results demonstrate that the proposed PolypSeg achieves much better segmentation results than state-of-the-art methods with a much faster speed.