The Iraqi army has stormed into Fallujah, capturing a police station inside the city limits, as it aims to retake one of the main strongholds of Islamic State militants.
Explosions and gunfire were ripping through Naimiya, a district of Fallujah on its southern outskirts, a Reuters TV crew at the city's edge said.
An elite military unit, the Rapid Response Team, seized the district's police station at midday, state television reported.
The battle for Fallujah is shaping up to be one of the biggest ever fought against Islamic State, in the city where US forces waged the heaviest battles of their 2003-2011 occupation against the Sunni Muslim militant group's precursors.
Fallujah is Islamic State's closest bastion to Baghdad, and believed to be the base from which the group has plotted an escalating campaign of suicide bombings against Shi'ite civilians and Government targets inside the capital.
As Government forces pressed their onslaught, suicide bombers driving a car and a motorcycle and another bomb planted in a car killed more than 20 people and injured more than 50 in three districts of Baghdad, police and medical sources said.
Separately, Kurdish security forces announced advances against Islamic State in northern Iraq, capturing villages from militants outside Mosul, the biggest city under militant control.
The Iraqi army launched its operation to recover Fallujah a week ago, first by tightening a six-month-old siege around the city, 50km west of Baghdad.
Fallujah, in the heartland of Sunni Muslim tribes who resent the Shi'ite-led Government in Baghdad, was the first Iraqi city to fall to Islamic State in January 2014. Months later, the group overran wide areas of the north and west of Iraq, declaring a caliphate including parts of neighbouring Syria.
On Monday, army units advanced to the city's southern entrance, "steadily advancing" under air cover from a US-led coalition helping to fight against the militants, according to a military statement read out on state TV.
Fallujah has been a bastion of the Sunni insurgency that fought both the US occupation of Iraq and the Shi'ite-led Baghdad Government that took over after the fall of dictator Saddam Hussein, a Sunni, in 2003.
The latest offensive is causing alarm among international aid organisations over the humanitarian situation in the city, where more than 50,000 civilians remain trapped with limited access to water, food and health care.
Fallujah is the second-largest Iraqi city still under control of the militants, after Mosul, their de facto capital in the north that had a pre-war population of about two million.
It would be the third major city in Iraq recaptured by the Government after Saddam's home town Tikrit and Ramadi, the capital of Iraq's vast western Anbar province.
Fallujah is also in Anbar, located between Ramadi and Baghdad, and capturing it would give the Government control of the major population centres of the Euphrates River valley west of the capital for the first time in more than two years.