In a message posted on Facebook, the airport says: "Following the attacks, there will be no flights into and out of Brussels Airport today.
"As soon as we have access to the terminal building, we can assess the damage. Later today we will assess when the operations can be resumed.
"Passengers are requested to contact their airline for further instructions.
"Brussels Airport extends its deepest sympathy to the friends and relatives of the victims of the explosions in the departures hall yesterday morning."
Belgium, Brussels and indeed much of Europe remains on high alert today after yesterday's atrocities which killed at least 34 people and injured hundreds more.