Some of the newer ships have sewage treatment plants on board. The purified water is dumped overboard almost anyplace; some local regulations require the purified water to be dumped in open water at least a mile from shore. The sludge is burned in either on board incinerators or on shore. Oasis of the Seas has its own sewage treatment plant and reuses the waste water.
International regulations require older ships to dump sewage at least 12 miles off shore. Some hold the sewage in tanks and dispose of it at shore facilities. At least that's what they're supposed to do. There are quite a few lawsuits against various cruise lines for illegal dumping.
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