More than 100 people were injured in the blast, which reports suggest was the work of a suicide bomber. Russia's chief investigator said the explosion was the work of terrorists, and President Dmitry Medvedev said the attackers would be tracked down. The airport is 40km (25 miles) south-east of the city centre.
In the aftermath of the blasts, Mr Medvedev ordered increased security across Russia's capital and at other airports, and called an emergency meeting with top officials.
He also postponed his planned departure for this week's World Economic Forum at Davos.
Reports suggest Monday's blast hit the baggage reclaim area of the airport's international arrivals hall.
A RIA Novosti news agency correspondent at the scene reported that a lot of smoke could be seen in the area, and there was a smell of burning.
The UK's Foreign Office said it was checking if any Britons were involved. Continue reading the main story
Terror attacks in Russia• Oct 2010 - Six people killed as militants storm parliament in Chechnya, North Caucasus
• Mar 2010 - Suicide bombings at two Moscow metro stations kill 40 people; attack blamed on N Caucasus militants
• Nov 2009 - Bomb blast hits Moscow-St Petersburg luxury express train, killing 26; N Caucasus Islamist group claims responsibility
• Sept 2004 - Chechen rebels seize school in Beslan; 334 hostages, including many children, killed in ensuing battle
• Aug 2004 - Suicide bomber blows herself up at a Moscow metro station, killing 10
• Aug 2004 - Two Tupolev airliners blown up in mid-air by suicide bombers, killing 89 passengers and crew
Briton Mark Green, who was on a British Airways flight that landed at the airport before the explosion, told BBC News there were thousands of people gathered in the baggage collection area, baggage hall and queue for immigration at the time of the blast."We were walking out through the exit of the arrivals hall towards the car, and there was this almighty explosion, a huge bang, we didn't know it was an explosion at the time, and my colleague and I looked at each other and said 'Christ that sounds like a car bomb or something,' because the noise was, literally, it shook you," Mr Green told BBC News.
He described scenes of panic and how he gave a drink of water to a bloodied Russian man whose face was blackened with soot.
Last March the Russian capital's underground system was rocked by two female suicide bombers from Russia's volatile Dagestan region, who detonated their explosives on the busy metro system during rush hour, killing 40 people and injuring more than 80.
BBC security correspondent Gordon Correra said immediate suspicion about Monday's attack would fall on militants from the Caucasus region.