Eyewitnesses saw armed men in black, their heads covered in scarves, entering the Westgate shopping centre on Saturday afternoon.
A senior figure in the Somali militant group al-Shabab has told the BBC it carried out a deadly attack on a shopping centre in neighbouring Kenya.
The gunmen have been cornered but an unknown number of hostages are still trapped inside Nairobi’s Westgate shopping mall, officials say.
Kenyan President Uhuru Kenyatta said 39 people had been killed, including some of his family, and 150 were injured.
Al-Shabab said the attack was in response to Kenyan troops in Somalia.
There are about 4,000 Kenyan troops in the south of Somalia, where they have been fighting the militants since 2011.
On its Twitter feed, al-Shabab – which has links to al-Qaeda – said it was behind what it called the “Westgate spectacle”.
Hostages trapped
In his TV address, Mr Kenyatta said security forces were “in the process of neutralising the attackers and securing the mall”.
He went on: “We shall hunt down the perpetrators wherever they run to. We shall get to them and we shall punish them for this heinous crime.”
He said he had “personally lost family members in the Westgate attack”.