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.