Belga news agency reported at least 13 people had been killed and 35 others injured in the blasts.
Interior Minister Jan Jambon announced that Belgium's terror threat had been raised to its highest level, as witnesses told there had been shots and shouts in Arabic at the airport before the blasts hit the departure hall.
Federal police told Belga that they could confirm one dead, amid chaotic scenes at the airport on the northwest outskirts of Brussels.
Television images showed passengers fleeing chaotic scenes, with thick black smoke rising from the terminal building, where the windows had been shattered. Another blast was reported at the Maalbeek metro station.
Images on social media showed collapsed floor tiles littering the floor of the terminal hall.
The airport has been shut down until further notice, Eurocontrol, the European organisation for air navigation safety, confirmed on its website.
Public broadcaster RTBF said regional authorities had gone into emergency mode.
RTBF said the blasts at the airport on the northwest outskirts of Brussels hit shortly after 8:00 am (0700 GMT).
The blasts come days after the dramatic arrest in Brussels on Friday of Saleh Abdeslam, prime suspect in the Paris terror attacks that killed 130 people in November, after four months on the run.
There was no immediate confirmation of the cause of the blasts.
Europe's main stock markets retreated as the news broke, with London's benchmark FTSE 100 index dropping 0.6 per cent compared with Monday's close and Frankfurt's DAX 30 shedding 1.1 per cent.
The ceilings collapsed, he said, describing a smell of gunpowder at the scene.
With shock on their faces, Jean-Pierre Herman embraced his wife Tankrat Paui Tran, who he had just gone to collect from the airport after her flight from Thailand.
"My wife just arrived," Herman said. "I said hello, we took the elevator and in the elevator we heard the first bomb.
"The second exploded just when we got off. We ran away to an emergency exit. I think we are very lucky."
Charlotte McDonald-Gibson, a British journalist living in Brussels, told AFP there had been "total confusion" at the airport, where she was having breakfast.
"Suddenly staff rushed in and said we have to leave," she said.
"They rushed out and into the main terminal A departures building. Nobody knew what was going on.
"It was total confusion, people were just standing around wondering what was happening."
There was no immediate confirmation of the cause of the blasts.
British premier David Cameron tweeted that his country would do "everything we can to help," and announced that Britain's COBRA security committee would meet today.
"The fight against this evil requires vital international cooperation," he added.
The blasts come as Abdeslam, Europe's most wanted man, remains in a high-security prison in Belgium following his arrest last week in the gritty Brussels neighbourhood of Molenbeek, just around the corner from his family home.
Belgium's Foreign Minister Didier Reynders said at the weekend that Abdeslam -- believed to have played a key logistical role in the carnage in Paris -- had been planning some sort of new attack.
"My wife just arrived," Herman said. "I said hello, we took the elevator and in the elevator we heard the first bomb.
"When we came out of the elevator at that moment the second bomb exploded and then we saw doors flying, (the) glass ceiling come down and smoke."
An AFP correspondent said roads to the airport had been blocked and trains halted.
Charlotte McDonald-Gibson, a British journalist living in Brussels, told AFP there had been "total confusion" at the airport, where she was having breakfast.
"They rushed out and into the main terminal A departures building. Nobody knew what was going on.
"It was total confusion, people were just standing around wondering what was happening.
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