The nationals of 109 countries and territories, including all the usual suspects, will receive a visa on arrival valid for 30 to 90 days; see the official Hi Korea site for the latest details. Rules for visiting only Jeju are even more lenient, allowing in everybody except citizens of 11 countries. Don't overstay, even by a single day — this incurs heavy fines and possible jail time, and you'll probably be banned from re-entering.
Military personnel travelling under the SOFA for South Korea are not required to possess a passport for entry, provided they hold a copy of their travel orders and a military ID. On the other hand, dependents must hold a passport and A-3 visa for entry.
You might need to prove to immigration officials that you have adequate means of support and accommodation upon entering Korea and that you have return or onward tickets.
Entry requirements for South Korea do change occassionally so we do advise checking with the embassy first.
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