US military deaths in Iraq are said to have fallen to their lowest monthly level for four years, after about 20 soldiers were reported killed in May.
The figures for Iraqi civilian deaths vary according to different sources, but have also dropped.
Most accounts put them at about 530 - or about half the levels seen in March and April.
Meanwhile, Australia has begun withdrawing its contingent of about 500 combat troops from Iraq.
The pullout honours a pledge made by the Australian Prime Minister, Kevin Rudd, when he was elected last November.
The Australian troops had been mainly playing what they call an "overwatch" role, assisting Iraqi forces.
The BBC's Jim Muir in Baghdad says one reason for the reduction in US military deaths was the ceasefire in early May, which stopped fighting in the Sadr City district of the Iraqi capital.
Iraqi and American troops had been engaged in an offensive there against Shia militiamen of the Mehdi Army, loyal to the cleric Moqtada al-Sadr.
But our correspondent adds that the overall trend of the violence since late last summer has been downwards.
He says the US troop "surge" was clearly a big factor, as was the trend within the Sunni community to turn against al-Qaeda and other Islamic militant groups.
The withdrawal of the additional US troops brought in last year for the surge
is expected to be completed by July.
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