BAGHDAD — David Petraeus, the American general who presided over Iraq's pullback from the brink of all-out civil war, relinquished his command today to Gen. Ray Odierno under a cascade of official thank-yous.

In an elaborate ceremony in a marble-lined rotunda of a former Saddam Hussein palace on the outskirts of the capital, Petraeus handed off to Odierno the responsibility for leading U.S. and coalition forces at a stage in the still-unpopular war that appears far more hopeful than when Petraeus assumed control 20 months ago.

Petraeus leaves behind a heavy dose of caution, reflected in his recommendation to President Bush that he maintain 15 combat brigades in Iraq through the end of the year instead of pulling out one or two, as many had expected.

And despite the security gains, insurgents retain the ability to carry out devastating attacks. On Monday evening, a female suicide bomber blew herself up among a group of police officers northeast of Baghdad, killing at least 22 people. Hours earlier, car bombs in the capital killed 13 people.

Defense Secretary Robert Gates presided over today's ceremony, and he praised Petraeus and Odierno for their accomplishments together in 2007, when Odierno served as the No. 2 U.S. commander and a revised U.S. strategy began to pay dividends.

This is the 3rd Iraq tour for Odierno, who told the gathering that while much remains for the U.S. military to accomplish here, the Iraqis must take charge. "This struggle is theirs to win," he said.

Petraeus' next assignment will be as commander of U.S. Central Command, with broader responsibilities. From his headquarters in Tampa, Fla., he will oversee U.S. military involvement across the Middle East, including Iraq, as well as Afghanistan, Pakistan and other Central Asian nations. He takes up that post in late October.

Petraeus and other military leaders regularly warn that security gains in Iraq are reversible and need continued U.S. attention — a point underscored by persistent bombings that bear the hallmarks of Sunni insurgents.