Outlook Mobile App Not Syncing

This issue is typically due to a synchronization error between the Outlook mobile application and the Office 365 server, which is quite easy to fix:

Open the Outlook Mobile app on your device, and click on the "three stacked bars" icon in the top, left-hand-corner. This will bring up your folders menu.
At the bottom you should see a 'gear' icon, which will take you to the settings page.
Under accounts, tap your email address and the app will take you to an Account Info screen.
Select the Reset Account option at the bottom.

This will force Outlook to re-sync and re-connect everything from your device, and establish a new connection to the server.