Office 365 is used for email, Sharepoint Online, OneDrive for Business and Skype. If you suspect a service outage and want to check Microsoft's notification site,
Visit: http://status.office365.com/ (This link is provided by Microsoft.com)
After a mailbox is moved to Office 365 your Outlook client should be restarted.
Upon restart it should prompt you for yoru username and password. Enter your email address; email password; and check the 'Remember my Credentials' checkbox.