Kenya's President Uhuru Kenyatta says the country is united and strong in adversity after a deadly attack on a Nairobi shopping centre on Saturday.
The Kenyan Red Cross says 68 people are now confirmed dead, after the bodies of nine hostages were recovered on Sunday.
In the past few hours, journalists outside a security cordon have reported gunfire and a large explosion.
Between 10 and 15 attackers - thought to be militants from the Somali al-Shabab movement - are still inside.
Some civilians are still trapped, either as hostages or in hiding.
At about 20:30 GMT on Sunday, the Kenyan Defence Forces (KDF) said on Twitter that most of the hostages had been rescued and security forces had taken control of most of the building.
Four KDF personnel were injured and taken to hospital.
"The criminals are now located in one place within the building," Mr Kenyatta said at a news conference on Sunday.
"With the professionals on site, we have as good a chance to neutralise the terrorists as we could hope for."
He thanked those who had helped with rescue and relief efforts, and asked other countries not to issue travel advisories against visiting Kenya.
Mr Kenyatta's nephew and his fiancee were among the dead, the president said.
The UK Foreign Office has confirmed that three Britons have been killed, and says the number is likely to rise.
French, Chinese, Ghanaian, Dutch, South African, Indian and Canadian nationals are also among the foreigners confirmed killed, along with a dual Australian-British national.
British Prime Minister David Cameron called it "an absolutely sickening and despicable attack of appalling brutality".
The Somali militant group al-Shabab says it carried out the attack in response to Kenyan military operations in Somalia.
There are about 4,000 Kenyan troops in the south of Somalia, where they have been fighting the militants since 2011.
There is a heavy military presence both in and around the shopping centre.
Late on Sunday afternoon, a police helicopter and another with military camouflage swept low over the shopping centre.
There are reports that the gunmen are currently holed up in a supermarket. Mr Kenyatta said women were reported to be among the attackers but said this was unconfirmed.
BBC security correspondent Frank Gardner says a security source told him that at least one of the attackers was a woman who appeared to have a leadership role.
'Watching and monitoring'
On Sunday afternoon, Kenyan officials said "major operations" were under way.
The BBC's Will Ross at the scene said it would be extremely difficult for the military to do a quick raid on the building because of all the people inside.
Al-Shabab has claimed there are at least 36 hostages, but this cannot be independently confirmed.
Our correspondent says the full extent of the attack will not be known until the military is back in control.
People continued to escape from the building on Sunday.
Cecile Ndwiga got out on Sunday morning, saying she had been hiding under a vehicle in the basement car park but could not leave earlier because "the shootout was all over - left, right".
The authorities have asked journalists to exercise caution when reporting military developments because the gunmen might be monitoring the media.
Thousands of Kenyans have responded to appeals for blood donations.
The attack began at about 12:00 local time (09:00 GMT) on Saturday, when the militants entered the Westgate centre, throwing grenades and firing automatic weapons. A children's day was being held at the time - children are among those reported killed.
Witnesses report seeing many bodies strewn round tables of unfinished fast food - with pop music left playing in the background.
Some witnesses said the militants told Muslims to leave and said non-Muslims would be targeted.
"They came and said: 'If you are Muslim, stand up. We've come to rescue you'," said Elijah Lamau.
He said the Muslims left with their hands up, and then the gunmen shot two people.
US President Barack Obama called President Kenyatta on Sunday to express condolences and reiterate "US support for Kenya's efforts to bring the perpetrators of the attack to justice".
The wife of an American working for the US Agency for International Development was killed, US officials said.
Prominent Ghanaian poet Kofi Awoonor - who was attending a literary festival in Nairobi - also died, as did a Chinese woman.
Security experts are reported to have warned that the Israeli-owned complex was in danger of being subjected to a terror attack.
Al-Shabab, which is part of the al-Qaeda network, has repeatedly threatened attacks on Kenyan soil if Nairobi did not pull its troops out of Somalia.
This is one of the worst incidents in Kenya since the attack on the US embassy in August 1998.