The U.S. airline said the problems were down to a power outage in Atlanta overnight and that its information technology team was working to resolve the problem.
Flights scheduled for departure were not taking off, but those already in the air were operating normally, Atlanta-based Delta said in a statement, telling customers to expect "large-scale cancellations".
The problems also meant flight information was not showing correctly on Delta's website or on airport information boards, it said.
Delta operates 5,000 departures a day and is a member of the SkyTeam alliance alongside airlines including Air France-KLM.
It also partners for transatlantic flights with Virgin Atlantic [VA.UL], which said its flights were operating normally but cautioned that passengers should check tickets in case their flight was due to be operated by Delta as part of a code share agreement.
"A power outage in Atlanta, which began at approximately 2:30 a.m. ET (0630 GMT), has impacted Delta computer systems and operations worldwide, resulting in flight delays and cancellations today," said Delta, the world's second-largest airline measured by revenue passenger kilometres flown.
"Our systems are down everywhere. Hopefully it won't be much longer," the airline said on Twitter earlier on Monday.
Passengers stuck in check-in queues in airports across the world, or on board planes waiting to depart took to Twitter to share photos and frustration at the delays.
"1 hr.+ lines @HeathrowAirport for @Delta due to system outage," tweeted user @MITJAKE with a picture of passengers waiting to check in.
The glitch follows several high-profile computer problems faced by U.S. airlines in the past year.
They included budget carrier Southwest Airlines Co, which had to halt departures last month after a technical outage, while American Airlines had to suspend flights from three of its hubs last September after technical problems.
Industry consultants say airlines face an increasing risk from computer disruptions as they automate more of their operations, distribute boarding passes on smartphones and fit their planes with Wifi.