Customers impacted by the iCloud outage who attempt to restore an iOS device using a backup are seeing the process hang while in progress, with the restore failing to complete. iCloud backups can still be made from iOS devices, so data is safe, but affected users will not be able to restore from backups until Apple's servers are back up. In some cases, existing iCloud backups are also not showing up on new devices.
Apple employees have been telling customers to wait it out and set up recently purchased iPhones and iPads as new devices rather than restoring from an existing backup.
Apple's iCloud Backup service has been experiencing issues since 8:00 a.m. yesterday morning, and it is not clear when a fix will be implemented. We'll update this post when the problem is resolved.