The San Francisco-based company acknowledged the problem, saying in a statement that its engineers are "currently working to resolve the issue," although it didn't go into any further detail.
Visitors to the site were greeted with a half-formed message partially in code saying that "Twitter is currently down."
The fields where a reason for the outage and a deadline for restoring service were apparently meant to go were filled with computer code.
Sluggishness or outages were reported from countries in North America, Europe, Asia, Latin America, the Middle East, and Africa.
Some users were apparently able to post updates - known as "tweets" - through their phones or third-party applications. Updates about the Olympic torch - which is currently wending its way through central London - were still pouring in, albeit far more slowly than earlier in the day.
About an hour after issues were first spotted, service appeared to be restored in parts of the U.S. and Asia, but users elsewhere continued to report problems.
The Olympics are expected to bring an unprecedented surge of activity by sports fans on social networking sites such as Twitter.
At the recent European Championship final, users fired off more than 15,000 tweets per second, setting a sports-related record for the site.
Social media users were already complaining about an earlier outage that affected Google's chatting services. The Mountainview, Calif.-based company said Thursday morning that the majority of users were seeing error messages and unable to use its Google Talk service. The issues were resolved five hours later.