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In this paper, we present a deep-learning-based framework for audio-visual speech inpainting, i.e., the task of restoring the missing parts of an acoustic speech signal from reliable audio context and uncorrupted visual information. Recent work focuses solely on audio-only methods and generally aims at inpainting music signals, which show highly different structure than speech. Instead, we inpaint speech signals with gaps ranging from 100 ms to 1600 ms to investigate the contribution that vision can provide for gaps of different duration. We also experiment with a multi-task learning approach where a phone recognition task is learned together with speech inpainting. Results show that the performance of audio-only speech inpainting approaches degrades rapidly when gaps get large, while the proposed audio-visual approach is able to plausibly restore missing information. In addition, we show that multi-task learning is effective, although the largest contribution to performance comes from vision.