Let`s start off with something simple and silly.
A colleague came over to me with a dilemma: A user, who was just upgraded to Office 365, had not received any meeting invites in his inbox since the upgrade. Instead, the invites went straight to the user`s calendar as tentative. The user being totally unaware of these invites, was missing meetings, which caused a whole bunch of consternation on the part of the user.
We tried everything. Checked Outlook settings, redid the mail profile, and sent a few test meetings; still nothing.
In the end it was a simple thing. After a couple of hours I figured it out. The user sorted his/her email by type. This meant that the user would only see regular emails. After we sorted the emails by date the issue was resolved.