Hundreds from around Europe have commented on the website Downdetector that they have been affected by the problem - many since Monday morning.
One common issue seems to be that sent emails remain in the drafts folder and are not being delivered to recipients.
On its website, Microsoft says the service dropped "unexpectedly" and it is working on a fix.
Not all account holders are affected.
"Intermittent connectivity is affecting customers in some European countries, which we are working to resolve as soon as possible," said a Microsoft representative.
Outlook incorporates Hotmail and Windows Live Hotmail accounts.
On its service status page, Microsoft is currently saying that an "alternate infrastructure" is being used while the service is restored.
"We've identified that a subset of infrastructure was unable to process requests as expected, which caused general service availability to drop unexpectedly," it says.
"We've redirected requests to alternate infrastructure to restore service, and we're monitoring the environment while connectivity recovers.
"Additionally, we're investigating an issue in which users are unable to send email messages."
Photo copyright MICROSOFT