Anyone who is using Outlook 2013 or Outlook 2016 to connect to their Outlook.com account who has not removed the Outlook.com EAS account from their profile and added it back using auto account setup, will need to make the change ASAP. EAS accounts will stop working in the near future. Microsoft is in the process of blocking Outlook desktop as an Exchange Active Sync (EAS) client for Outlook.com users.
This protocol was blocked for Office 365 business users against Exchange Online (and in on-premise Exchange Servers) because it offered a very poor user experience. EAS was designed to provide a good experience using the lightweight mail clients found on smartphones and not well suited to Outlook but it was enabled for Outlook.com accounts to make migration to the new server easier.
If you are using an outside address, you'll need to add an alias to your account in order to set it up in Outlook. Instructions are at Add an Alias to an Outlook.com Account.
However, due to issues that are amplified by having many people use Outlook EAS, Microsoft is shutting EAS access off for all Outlook desktop users. Smartphone apps are not affected, they will still be able to using EAS. Only Outlook desktop is affected.
Using the full Microsoft Exchange to connect is by far the best option, with full access to all Outlook and Exchange features available in consumer accounts. Some users switched back to EAS to avoid an Outlook.com bug where, under certain, specific conditions, all replies where sent using the Outlook.com account. These users will need to remove the EAS account from their profile and use auto account setup to add it back. (To work around the bug, users can set a pst file as default if they want to avoid selecting the correct account each time they reply.)
If an account is blocked before a user makes the switch, the account will stop syncing. When EAS is shut off, Microsoft will put an email into the mailbox telling the user to remove the account and reconnect to Outlook.com.
Switching to Microsoft Exchange account type is simple to do: after verifying any folders labeled 'this computer only' are empty, go to File, Account Settings, Email tab and Remove the Exchange ActiveSync account. (If it's the only account in the profile, create a new profile instead.)
Click New and type in your email address then click Next. The name and password fields can be left blank for Outlook.com and Office 365 Exchange accounts. Outlook will ask for your password if it doesn't find it in the Windows Credential Manager. Your name is picked up from your Microsoft account.
A reminder for Outlook 2007 users connecting to Outlook.com or Office 365 Exchange: you'll need to upgrade before October 31 2017 as Outlook 2007 will no longer connect using Microsoft Exchange account type. POP and IMAP will be supported but are email only, calendar and contacts will stop syncing.