You must have a valid passport to enter the Republic of Korea. U.S. citizens can enter the Republic of Korea without a visa for a stay of 90 days or less for tourism or temporary business purposes. If you are visiting the Republic of Korea for employment, for any profit-making reason, to teach English, or for stays longer than 90 days, you must get a visa at an ROK embassy or consulate prior to entering the Republic of Korea. In addition, if you plan to stay for longer than 90 days, you must apply for an Alien Registration Card.
Remember to check for updates at the US Department of State website.
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Source: US State Department