Gmail Backups
Backups are generally more important for POP accounts than for IMAP accounts.
To tell what type (IMAP or POP) of email account your Gmail is, log into your Gmail account and open the Settings page with the gear icon. Click on the
Forwarding and POP/IMAP tab & you can tell what sort of account you have there.
If your account is the POP type, then you definitely need to figure a backup plan. Here's a link to an article on ZDNet with nice coverage of the options:
The ultimate guide to Gmail backup | ZDNet
That article covers IMAP backups as well.
If you have an IMAP Gmail account, though = you should know that Google maintains backups of its servers. The default storage limit is 15 GB, and your non-deleted mail stays on the servers (unless you choose to delete it). However, if you have critical data stored in your email, you might prefer to have a local backup anyways.
I'd go with the Thunderbird option, since it is well-supported & constantly updated. Live Mail seems to be getting pushed out the door by Microsoft. If you happen to have a version of Microsoft Office that includes Outlook, you could use Outlook for the task.
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Note on the Gmail storage limit:
The current free storage for Gmail/Drive/Photos is 15 GB (the storage is shared between the three ... if Photos are set to "high quality", the the photo storage is unlimited, and does not count against the 15 GB free storage). So, if you don't use either Google Drive or Google Photos: you have 15 GB of storage as a limit for your Google email (and you can purchase more room, if necessary, or acquire more room through promotions ... ). Your mail is stored encrypted on their servers indefinitely, until you either delete them yourself or cancel your account.
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Windows 10 Mail
As of the last Windows 10 build (the December 8th "Redstone"), I haven't yet seen an option added to Windows 10 for local backups of the folders ... through the interface. Some of the mail is stored locally, of course, but the interface does not provide an easy way to get to these. It's possible that at some point in the future, Microsoft will add an export/backup feature to Windows 10 Mail - but I don't have a crystal ball for that one. You can still use the Windows 10 Mail as your main email app, should you prefer it - you just need to also sync to a "client" software like Thunderbird or Outlook for the local backup functions. For now, anyway!
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Windows Live Mail
Microsoft seems to be doing quite a lot to push users from Windows Live Mail - I can't imagine that the future for Live Mail will get any brighter. So I'd avoid that option. Seems "popular but doomed".
. . . Let us know if you have any questions, or need more details.