Belgian officials have confirmed casualties but have given no numbers. The cause of the explosions is unknown.
Another explosion has now been reported at the Maelbeek metro station close to the EU institutions. The whole metro system has been closed.
The explosions come four days after the capture in Brussels of Salah Abdeslam, the main suspect in the jihadist attacks in Paris in November 2015.
Belgium has now raised its terror threat to its highest level.
Emergency services are clearing the airport completely. Belgium’s Het Laatste Nieuws reports that more bombs have been found.
The airport is being evacuated and has been closed to flights.
Shots were fired and shouts in Arabic were heard before the two explosions, the Belga news agency reports.
Images on social media showed smoke rising from one of the buildings, amid reports of panic as people fled the airport.
Rail transport to the facility has been halted and people have been told not to come to the airport.
All flights have been diverted.
Other reports say at least one of the explosions was close to the American Airlines check-in area but this has not been confirmed.
The Belgian broadcaster RTBF quoted a witness as saying there were people injured or unconscious in the departure area, opposite the Sheraton hotel.
There are few reports yet about the metro blast.
Belgium’s Interior Minister Jan Jambon had said on March 21 that the country was on the highest level of alert for possible revenge attacks after the capture of Salah Abdeslam.
He told Belgian radio: “We know that stopping one cell can… push others into action. We are aware of it in this case.”
Zaventem airport is 7 miles north-east of Brussels and dealt with more than 23 million passengers in 2015.
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