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For POP, do as war17 says. For IMAP to download all your messages to your local machine (the messages are on your server by definition, so you don't have to worry about messages being deleted while you download), follow this: Tools - Account Settings [your IMAP account] - Offline and Diskspace - Select Folders for Offline use, and check off the folders whose messages you would like to make available offline (i.e. download to your local machine).
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