Email notifications not working

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We are not receiving any email notifications or form approval emails from any of our processes.  I have successfully registered an email address and turned on email notifications in the Configure Services section.  Also, I went into the Notifications Log and ran a test notification.  It said it was successul, but that doesn't even show up in the log and I still don't get an email.

Are we missing some other administrative task or role assigment or permissions that would allow emails to be sent?  We were able to get this to work in our workshop environment by simply registering an email address and turning on email notifications, but that didn't do the trick in our live environment.