US Marines said six of their people have been killed in the fighting, with another 14 injured.
There are also some Iraqi casualties.
Prisoners of war
Both sides are thought to have taken prisoners of war. Iraqi television showed interviews with American men and women they claim to have taken prisoner.
George Bush later appealed for them to be treated with respect, while British Prime Minister Tony Blair said it went against human rights rules to parade prisoners on television in that manner.
Bombed
This comes after one of the longest-running battles so far in the town of Nassiriya in southern Iraq.
Ground troops had to call for air strikes to help them beat around 500 Iraqi soldiers who were trying to stop them taking control of the town.
US aircraft also bombed parts of the port town of Umm Qasr, where another ground battle took place.
Other war updates:
Baghdad was heavily bombed again throughout Sunday. Explosions and fires were reported all around the city.
An RAF aircraft was shot down by mistake by a US missile. American military bosses said it was a "tragedy" and launched an investigation into why it happened.
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Iraq war: hour by hour updates
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