Iraqi special forces expect to encounter fierce resistance as they push into Fallujah in a bid to free the territory from Islamic State (IS) control.
The city 40 miles west of Baghdad has been under militant control longer than any other part of Iraq, and IS fighters have had more than two years to dig in.
Networks of tunnels like those found in other IS-held territory have already been discovered in the city's north-eastern outskirts.
The Iraqi troops, also known as the counter-terrorism forces, are leading the assault on Fallujah, slowly moving up from the southern edge in a column of armoured vehicles.
Their advance is expected to be slow because tens of thousands of civilians remain trapped in Fallujah and hidden bombs are believed to have been left throughout the city, according to special forces commanders at the scene.
They expect fierce resistance from the jihadis, who have nowhere to run.
"This is the decisive battle for us and for Daesh," said General Saad Harbiya head of Fallujah operations for the Iraqi army, using an Arabic acronym for IS.
The offensive, supported by air strikes from the US-led coalition, was launched a week ago.
In that time, other wings of Iraq's security forces have cleared the city's edges. Shia militia forces under the government umbrella of the Popular Mobilisation Forces and the federal police lead operations that have taken back 80% of the territory around Fallujah, according to Iraqi Major Dhia Thamir.
The predominantly Sunni city in Anbar province is one of the last major IS strongholds in Iraq.
The extremist group still controls territory in the north and west, as well as the second-largest city of Mosul.
Gen Harbiya said Fallujah "is like the Kaaba" for IS, referring to the most sacred Muslim site in the world in Mecca, Saudi Arabia.
The 500-700 IS fighters holed up in Fallujah are expected to be some of the group's best-trained, a special forces commander at the scene said.
The counter-terrorism forces started pushing into Fallujah from its southern edge at dawn, said Brigadier Haider al-Obeidi.
He described the fighting as "fierce," with IS deploying snipers and releasing a volley of mortar rounds on the Iraqi forces.
Humanitarian groups say that as the violence intensifies, their concerns for civilians trapped inside Fallujah mount.
"With every moment that passes, their need for safe exits becomes more critical," said Nasr Muflahi, the country director for the Norwegian Refugee Council, an international humanitarian group active in Anbar province.
In past operations, Iraq's Shia militia forces have been accused of committing abuses against civilians in majority Sunni towns and cities.
Sunni politicians already have accused the security forces of using indiscriminate force that has endangered the more than 50,000 civilians estimated to be still inside Fallujah.
Shia militia commanders have routinely rejected the accusations.
"The troops have been recommended to respect families and treat them gently," said Hadi al-Amiri, the Shia militia commander who also heads the Badr Organisation political party, while overseeing operations outside Fallujah.
IS extremists, meanwhile, claimed responsibility for a wave of bombings Monday in and around the capital that killed at least 24 people.
IS has been behind many of the recent deadly attacks, and the bombings show the group's enduring ability to launch operations despite territorial losses. Iraqi officials say the bombings are an attempt by the militants to distract the security forces' attention from the front lines.
"By launching such attacks, the militants aimed at thwarting our determination and resolution to continue with our victories in Fallujah," said Arkan Jabbar, a soldier manning a checkpoint in Baghdad not far from where one of the blasts hit.
The deadliest of the blasts took place in the northern, Shia-dominated Shaab neighbourhood of Baghdad, where a suicide bomber rammed his explosives-laden car into a checkpoint next to a commercial area, killing eight civilians and three soldiers.
The explosion also wounded up to 14 people, a police officer said.
A suicide car bomber struck an outdoor market in the town of Tarmiyah, about 31 miles north of Baghdad, killing seven civilians and three policemen, another police officer said, adding that 24 people were wounded in that bombing.
And in Baghdad's eastern Shia Sadr City district, a bomb on a motorcycle went off at a market, killing three and wounding 10, police said.
In an online statement, IS said it was responsible for the attacks, saying they targeted members of the Shia militias and a government office.