Religion of peace goes shopping ..
Prime Minister David Cameron called it "an absolutely sickening and despicable attack of appalling brutality".
Kenya's Red Cross says 68 people were confirmed dead after the attack.
Between 10 and 15 attackers and an unknown number of civilians are still in the building.
The Somali militant group al-Shabab says it carried out the attack on the upmarket Westgate shopping centre in response to Kenyan military operations in Somalia.
The group claimed that a 24-year-old man from London was one of the attackers. The al-Shabab Twitter account published a list of six people who took part in the attack before it was suspended.
"They do it in the name of terror, violence and extremism and their warped view of the world.
Stand-off
Al-Shabab has claimed there are at least 36 hostages, but this cannot be independently confirmed. There are also likely to be people hiding away from the attackers.
NAIROBI, Kenya — Seeking to end a siege that has left scores dead and shaken the nation, the Kenyan government said Sunday night that it was pressing an assault against Shabab militant attackers who had been holed up in a Nairobi shopping mall for more than a day.
“This will end tonight — our forces will prevail,” the police command center said in a Twitter post. “Kenyans are standing firm against aggression, and we will win.”
The Shabab, a militant group mostly based in neighboring Somalia, answered with messages of their own, including warning that “Kenyan forces who’ve just attempted a roof landing must know that they are jeopardizing the lives of all the hostages at #Westgate.”
Later, officials said that at least four members of the security forces had been wounded. But there were no other details about additional casualties on either side.
The attack on the mall deeply distressed Kenya, a nation that has grown in stature as a force against terrorism in East Africa. As the toll mounted — at least 68 were reported dead by late Sunday, with several people still unaccounted for — the potential for even greater loss of life seemed tangible.
The assault on Westgate was the deadliest terrorist attack in Kenya since the 1998 Qaeda truck bombing of the American Embassy in Nairobi that killed more than 200 people.