Operation Lightning began with hundreds of checkpoints set up across Baghdad, searching cars and people in an effort to halt insurgent attacks that have killed more than 700 people in the past month.
The operation, involving up to 40,000 thousand Iraqi soldiers and police, is strongly supported by US forces.
Iraqi security forces hope the crackdown will capture and deter insurgents around Baghdad and in other areas where attacks have become increasingly frequent.
Under the plan, the capital will be divided into two sectors, Karkh on the west bank of the Tigris river, and Risafa on the east.
Karkh will be divided into 15 sub-districts and Risafa into seven. Police and emergency personnel will operate 24 hours a day.
It was not known how long Operation Lightning would last.
But insurgents wasted no time defying the offensive, launching a series of coordinated attacks on Sunday.
Four car bombs in and around the capital killed 16 people, most of them
Nine soldiers taking part in Operation Lightning died when a suicide car bomb exploded at their roadblock just south of the capital.
Two other policemen were killed when a suicide car bomber targeted their patrol in south-western Baghdad.
In western Baghdad, a car bomb targeting police commandos killed three
people and wounded 20, an interior ministry source said, adding that police had then fought a firefight in the area.
An earlier suicide bombing near the oil ministry left two dead, while
violence elsewhere claimed the lives of a British soldier and seven Iraqis.
The British soldier died and four others were injured when his convoy came under attack in southern Iraq.
His death brought to 88 the number of British troops killed since the start of the Iraq war.
An Internet statement released in the name of al-Qaida in Iraq claimed responsibility for Sunday’s attacks, which it said were in response to Operation Lightning and the detention of Iraqis at Abu Ghraib prison.
The statement’s authenticity could not be verified.
British newspaper The Sunday Times reported Iraq’s most wanted man, insurgent leader Abu Musab al-Zarqawi, left the country after being wounded in a missile attack and was thought to be in Iran.
Tehran denied al-Zarqawi was in the country.