Images on social media from the scene showed smoke rising from one of the terminal buildings. The cause of the explosions is unknown.
Some reports from the scene speak of casualties but this has not been confirmed.
The blasts come four days after the capture in Brussels of Salah Abdeslam, the main suspect in the Paris attacks in November.
Belgian media say the airport is being evacuated. It has been closed to flights. Rail transport to the facility has been halted.
Social media reports from the scene speak of panic as people fled the airport buildings.
Images showed a swathe of shattered glass on one of the buildings.
Explosions at Brussels airport, train station
Other reports say at least one explosion was close to the American Airlines check-in area but again this has not been confirmed.
Belgian prosecutor saying that #Brussels airport explosion caused by suicide attack
— Anna Ahronheim (@AAhronheim) March 22, 2016
#BREAKING: Two loud explosions at #Zaventem airport in #Brussels pic.twitter.com/JFw9RGLjnh
— Anna Ahronheim (@AAhronheim) March 22, 2016
New footage outside #Maalbeek Metro in #Brussels shows victims being wheeled away by emergency services https://t.co/bdDNbA1ocM
— Sky News (@SkyNews) March 22, 2016
At least 13 people killed & 35 injured in #Brussels attacks - Belgian federal prosecutor https://t.co/u0Ub0PPknY pic.twitter.com/eSPbu0ishr
— BBC Breaking News (@BBCBreaking) March 22, 2016
Explosion at #Brussels Airport. Please follow instructions of Belgian security authorities https://t.co/EbXt9CpEG5 pic.twitter.com/HOqPhlZu3Q
— Foreign Office (FCO) (@foreignoffice) March 22, 2016
As we would expect after the horrendous and sickening events in #Brussels the police have a helo up. #POTN pic.twitter.com/xxCBo6Ubxx
— Planes on the net (@planesonthenet) March 22, 2016