While Microsoft might deem file shares to be old fashioned and prefer you to use SharePoint online, most businesses still use file shares to a greater or lesser extent.
Being able to replicate the files to Azure and back down to other sites is the aim of the newly-launched Azure File Sync.
Check out the video below for a short overview.
Take a look at Microsoft's blog post below for the full story. In the meantime note that, while NTFS permissions are synchronised to Azure and back down to other file servers (if present), the file permissions are not processed when accessing the Azure copy directly. This, however, is set to change in a future release.
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