Twin blasts hit Zaventem airport at 07:00 GMT, killing 11 and injuring 81, Belgium's health minister said.
Another explosion struck Maelbeek metro station an hour later. Twenty people were killed, Brussels mayor Yvan Mayeur said.
Belgium has now raised its terrorism threat to its highest level.
The attacks come four days after Salah Abdeslam, the main fugitive in the Paris attacks, was seized in Brussels.
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Thoughts go out to families and friends of the victims of this mornings attacks on Brussels. In the days to come we will no doubt seek answers as to why and how these events have come to pass.
International Response coming through
Belgium’s Interior Minister Jan Jambon has announced three days of national mourning, writes Oliver Milne.
Jambon is expected to join Belgian Prime Minister Charles Michel at Zaventem airport in the next few hours, where they will tour the scene of this morning’s atrocities.
Obama was briefed on the Brussels explosions Tuesday morning at at the U.S. Embassy in Havana, where he was staying on the third day of his Cuban visit, White House officials said.
He spoke to Belgian Prime Minister Charles Michel before the speech and offered the condolences of the American people, the White House said. Obama offered U.S. assistance in investigating the attacks.