Although there are small groups of Iraqi soldiers still fighting back, most have apparently escaped the town.
But not everyone is happy with the Kurds' success in Kirkuk.
Turkey - the country directly to the north of Iraq - says it is unacceptable that the Kurds are allowed to control the city.
They are worried that Kurdish people might try and build their own brand new country in the north of Iraq - something the Turkish government really doesn't want.
Kurdish commanders have told journalists that their soldiers will leave the city soon, making way for more US troops.
Other main points
A prominent Iraqi Shia Muslim leader, Abd al-Majid al-Khoei, and another cleric, Haydar Kilidar, are murdered in a mosque in the southern holy city of Najaf - and the White House strongly condemns the attack
US President George Bush and UK Prime Minister Tony Blair broadcast messages direct to the Iraqi people on a new TV station, Towards Freedom, set up by coalition forces
US and UK planes bomb Saddam Hussein's home city of Tikrit, about 145 kilometres (90 miles) north of Baghdad