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How do I backup Yahoo's message history?
Assuming you mean Yahoo Instant Messenger (YIM), then I believe it's just the contents of a specific directory.
I'd look at: C:\Program Files\Yahoo!\Messenger\profiles\name\Archive, where name is replaced by the screen name you're using. Remember that Yahoo's archiving needs to be turned on (it's off by default) and that also by default YIM will automatically delete messages after 10 days.