Your contacts and calendars can take care of themselves
Contact and calendar management can be tricky nowadays, what with competing data on phones, Facebook, Gmail, and the like. The good news is that you may already have a backup system in place (or at least available) without realizing it.
For example, if you're an iPhone user, you can enable iCloud (via Settings, iCloud) for Contacts and Calendars, and they're all set: All your addresses and appointments will stay synced with your iCloud account (which you can also access via a Web browser in case your phone gets lost, stolen, or broken).
iCloud (pictured) and Google will automatically back up your contacts and calendar to preserve them in the event you lose your device.
Likewise, Android devices automatically sync with Google Contacts and Google Calendar, effectively giving you an automated backup that you can restore or view online as needed.
Store your music in a digital locker
One radical way to keep a "backup" of your music library is to ditch it altogether, instead relying on all-you-can-stream services like Rdio and Spotify. Of course, most of us still like to keep songs on our PCs, phones, and tablets, in which case it once again makes sense to look to the cloud for backup purposes.
For iTunes users, Apple's Match service can store your music library in iCloud, where it's available for streaming or download (should you need to restore it to your PC). However, it costs $25 annually.
Google Play lets you upload your own songs for free and make them available to yourself on any Internet-enabled device.
A cheaper option: Google Play, which lets you upload as many as 20,000 songs (including any you might have in iTunes) to your digital locker and make that music available anywhere. Its Windows client automatically syncs new music to your account. There's a download option in case you need to restore your library. Price: zero.
If you still need to back up email, use Mailstore
One of the big benefits to a Web-based email service like Gmail, Outlook.com, or Yahoo is that your mail already lives online. Because nothing is stored on your PC, there's nothing to back up.
However, if you're using a mail client such as Outlook, Windows Live Mail, or Thunderbird to retrieve and manage your mail, then it's a good idea to make a backup. That way, should calamity strike, you can restore your entire email archive, no harm done.
If you still use a desktop mail client, use MailStore to back up your inbox for free.
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