By Rageh Omaar and Paul Wood
BBC correspondents in Baghdad
Americans control much of the city
The Americans pretty much control Baghdad, with just mopping-up operations going on, dealing with small pockets of resistance; but these operations could all take days, even weeks longer.
The military phase is almost over, although there is no question that on the west side of the city, some Iraqis are continuing to put up a fight.
There certainly has not been anything like the celebrations that there were on the east side of the city on Wednesday.
The people here don't have electricity to see the television pictures of that night, but nonetheless their impact throughout the Arab world cannot be overstated.
To many, Saddam Hussein had represented rare Arab defiance of US power, his fate is a warning to other unelected Arab rulers.
The shattered bronze head of Saddam Hussein's statue was still being dragged around Baghdad on Thursday.
I'm going to exercise my right of free speech for the first time in my life - we want you out of here as soon as possible
One of the things that staggered the coalition commanders was the amount of ammunition lying around on the streets.
It is as if the Republican Guard came up with the idea of dumping ammunition, block after block, so that if they had to move, there would be ammunition waiting for them.
Marines say that the US flag draped over Saddam Hussein's statue was the flag that was flying over the Pentagon on 11 September 2001.
Looting has continued throughout the city
It was later replaced with the Iraqi flag after the people shouted for it.
One of my close Iraqi friends went up to an American marine and said to him: "I'm going to exercise my right of free speech for the first time in my life - we want you out of here as soon as possible."
The US does not have enough troops on the ground in Baghdad to maintain civil order.
But there is going to be a political urgency in the next few days for coalition forces to calm things down and stop the looting.
Baghdad on Wednesday night was a city without law, order or electricity. Intermittent small arms fire was heard as people fought off robbers from their barricaded homes.
The US forces were doing nothing to stop the looting, but the next big challenge for the coalition troops in Baghdad will be to convert what is a relatively small war fighting force into a policing operation.
The problem will become acute if, as many people expect here, the looting spills over into private homes and businesses.
And the real problem remains the die-hard Fedayeen who are still shooting, bombing and killing civilians and marines.