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In this paper we propose a generative adversarial network (GAN) framework to enhance the perceptual quality of compressed videos. Our framework includes attention and adaptation to different quantization parameters (QPs) in a single model. The attention module exploits global receptive fields that can capture and align long-range correlations between consecutive frames, which can be beneficial for enhancing perceptual quality of videos. The frame to be enhanced is fed into the deep network together with its neighboring frames, and in the first stage features at different depths are extracted. Then extracted features are fed into attention blocks to explore global temporal correlations, followed by a series of upsampling and convolution layers. Finally, the resulting features are processed by the QP-conditional adaptation module which leverages the corresponding QP information. In this way, a single model can be used to enhance adaptively to various QPs without requiring multiple models specific for every QP value, while having similar performance. Experimental results demonstrate the superior performance of the proposed PeQuENet compared with the state-of-the-art compressed video quality enhancement algorithms.