Apple has confirmed its app stores and music and radio services are disrupted.
Users in the UK were among those affected by the "outage", which began at about 13:00 on Tuesday.
On website Downdetector, which monitors internet issues, users from as far afield as Australia, the US, Vietnam and Qatar were also reporting issues.
Apple's System Status page says: "We are investigating and will update the status as more information becomes available."
According to Apple news site 9to5Mac, the red triangle denoting outage is almost never seen on its status page.
"Using the red triangle is indicative of a very significant problem," it said.
The problems come just after the trial versions of iOS 13 and macOS Catalina were made available to developers and days after Google Cloud services were affected by a major blackout, which brought down numerous websites and services.
That problem also affected many of Apple's iCloud products, including iMessage and photos.
Apple uses Google Cloud as the backbone for some of its products.