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Satellite images hold great promise for continuous environmental monitoring and earth observation. Occlusions cast by clouds, however, can severely limit coverage, making ground information extraction more difficult. Existing pipelines typically perform cloud removal with simple temporal composites and hand-crafted filters. In contrast, we cast the problem of cloud removal as a conditional image synthesis challenge, and we propose a trainable spatiotemporal generator network (STGAN) to remove clouds. We train our model on a new large-scale spatiotemporal dataset that we construct, containing 97640 image pairs covering all continents. We demonstrate experimentally that the proposed STGAN model outperforms standard models and can generate realistic cloud-free images with high PSNR and SSIM values across a variety of atmospheric conditions, leading to improved performance in downstream tasks such as land cover classification.