We introduce translation error correction (TEC), the task of automatically correcting human-generated translations. Imperfections in machine translations (MT) have long motivated systems for improving translations post-hoc with automatic post-editing. In contrast, little attention has been devoted to the problem of automatically correcting human translations, despite the intuition that humans make distinct errors that machines would be well-suited to assist with, from typos to inconsistencies in translation conventions. To investigate this, we build and release the Aced corpus with three TEC datasets (available at: github.com/lilt/tec). We show that human errors in TEC exhibit a more diverse range of errors and far fewer translation fluency errors than the MT errors in automatic post-editing datasets, suggesting the need for dedicated TEC models that are specialized to correct human errors. We show that pre-training instead on synthetic errors based on human errors improves TEC F-score by as much as 5.1 points. We conducted a human-in-the-loop user study with nine professional translation editors and found that the assistance of our TEC system led them to produce significantly higher quality revised translations.