Another explosion struck Maelbeek metro station near EU headquarters an hour later, leaving about 20 people dead.
Brussels police have issued a wanted notice for a man seen pushing a luggage trolley through the airport. He was pictured in CCTV footage with two other suspects who are believed to have died in the blasts.
The Islamic State (IS) group said it was behind the attacks in a statement issued on the IS-linked Amaq agency.
Belgium has raised its terrorism alert to its highest level. Three days of national mourning have been declared.
Not only is Brussels a high-profile target for Islamists, Belgium has struggled with Islamist groups for years and hundreds of its citizens have been lured into fighting for IS in Syria and Iraq.
Several of the bombers and gunmen who targeted Paris last November, killing 130 people, had been living in Molenbeek. The main suspect not to die in the Paris attacks, Salah Abdeslam, returned to Belgium the day afterwards and managed to evade police until 18 March. He and an accomplice were captured alive, again in Molenbeek.
Many Belgians were expecting a response from jihadists. “I had certainly expected something else would take place, but not that it would happen on this scale,” says Belgian jihadism expert Pieter Van Ostaeyen.