Most video super-resolution methods focus on restoring high-resolution video frames from low-resolution videos without taking into account compression. However, most videos on the web or mobile devices are compressed, and the compression can be severe when the bandwidth is limited. In this paper, we propose a new compression-informed video super-resolution model to restore high-resolution content without introducing artifacts caused by compression. The proposed model consists of three modules for video super-resolution: bi-directional recurrent warping, detail-preserving flow estimation, and Laplacian enhancement. All these three modules are used to deal with compression properties such as the location of the intra-frames in the input and smoothness in the output frames. For thorough performance evaluation, we conducted extensive experiments on standard datasets with a wide range of compression rates, covering many real video use cases. We showed that our method not only recovers high-resolution content on uncompressed frames from the widely-used benchmark datasets, but also achieves state-of-the-art performance in super-resolving compressed videos based on numerous quantitative metrics. We also evaluated the proposed method by simulating streaming from YouTube to demonstrate its effectiveness and robustness. The source codes and trained models are available at https://github.com/google-research/google-research/tree/master/comisr.