“Burned people are running about,” eyewitness Andrei said shortly after the blast, “they are carrying pieces of flesh on stretchers,” he added.
“Dozens of people are being dragged out on stretches, on trolleys,” he told City FM radio.
"I was 10 metres away from the explosion. Suddenly there was a flash and then very thick smoke appeared. Everyone panicked because we feared that there would be a second explosion," eyewitness Artyom Zhilenkov told the AFP news agency.
He said many of the victims were taxi drivers gathered outside the arrivals hall.
Before the explosion jets from Dusseldorf, Odessa and London had all landed at the airport. A British citizen and other foreigners are among the fatalities, according to a source quoted by the Russian Interfax news agency.
Investigators said they believe they have found the remains of the suicide bomber who was a resident of the Muslim northern Caucasus region.
"We have found the head of a man of Arab appearance who is about 30 to 35 years old," Interfax reported an unnamed security source as saying. "He presumably set off the explosive device," the official added.
Vladmir Markin, a spokesperson for the Russian investigative committee, said an inquiry had been opened into what he called an “act of terror”.
Medvedev has vowed to track down those responsible and ordered heightened security measures. He also postponed his trip to the World Economics Forum in Davos. He was due to fly to Switzerland on Tuesday.
"It is necessary to introduce a special regime in all airports and transportation hubs," Medvedev said in televised remarks. "What happened indicates that far from all the laws that need to be working are being used correctly," he said.
There was international condemnation of the attack.
United Nations chief Ban Ki-moon was "appalled" by the "deplorable and unjustifiable act", a UN spokesperson said. Ban conveyed his "deepest condolences".
United States President Barack Obama called it "outrageous" and offered "any assistance that the government of Russia needs", in comments quoted by White House spokesperson Robert Gibbs.
European Union President Herman Van Rompuy sent a message of "solidarity" in his statement. He called for "those behind the suicide bomb to be tracked down and punished".
Moscow has seen repeated attacks over the last few years. Explosions carried out by female suicide bombers killed 40 people and wounded more than 100 in March last year. In 2006 a bomb in a crowded market killed 14 people and injured 47.