Drag and Drop Emails to Your Local Device:

There's a few options for saving multiple emails from Outlook. The simplest option is to select those emails you want to save and drag them to a folder in Windows Explorer for Windows users, or Finder for those using a Mac. Your emails will be saved as individual .msg files with the subject as the filename. Any attachments in your emails will also be saved.

If you choose this option, then be aware that you can only re-open your saved emails in Outlook, as the .msg format is proprietary to Outlook. In order to open these emails with a different program, you will need to use an MSG converter or viewer software. Anyone using Office 2011 on a Mac however can re-save these emails as .eml files, which can be opened on other email clients.

Export All of Your Outlook Folders:

If you would like to fully back-up all of your email folders from Outlook, then you can do so by using Outlook's export funtion. For Outlook users, this can be found under 'File' > 'Options' > 'Advanced'. Your folders will be saved as a .pst file, meaning they can be opened with Outlook on other computers.

Save Individual Emails in Outlook:

You can save individual emails in Outlook in a universal format like text or HTML, by going to 'File', 'Save As'. You will need to save any attachments in your email separately, otherwise they will be deleted.

Additional info on how to back-up your emails can be found on the official Outlook page about the matter.