Most, if not all, forms of ellipsis (e.g., 'so does Mary') are similar to reading comprehension questions ('what does Mary do'), in that in order to resolve them, we need to identify an appropriate text span in the preceding discourse. We present a strong baseline for English ellipsis resolution that exploits this similarity by relying on architectures developed for machine reading comprehension. We present both single-task transfer learning models and joint models, trained on machine reading comprehension and coreference resolution datasets, clearly outperforming the current (from 2016-18) state of the art for Sluice Ellipsis (from 0.67 to 0.86 F1) and Verb Phrase Ellipsis (from 0.65 to 0.79 F1).