Reporters on the ground and military spokesmen said the advancing U.S. and Iraqi troops were met by rifle fire, mortars and rocket propelled grenades as they moved through the city "street by street" and "house by house."
According to reports from embedded reporters, U.S. and Iraqi troops have surrounded Fallujah's al-Hidra mosque, which the military said the insurgents were using to store weapons, and have advanced past the city's central thoroughfare.
Fighting in the mainly urban area has created challenges for coalition forces. An original assault on the city in April was called off after hundreds of Iraqi civilians were reportedly killed or wounded. Reporters on the ground have said most civilians have fled the city but a few remain.
"Iraq forces are leading the attack through culturally sensitive areas," Maj. Gen. Abdul Qader Mohan, the commander of Iraqi forces in Fallujah, said. "Areas with schools, hospitals and mosques (will be) under the operational control of the Iraqi army."
Lt. Gen. Thomas Metz, commander of multinational forces in Iraq, said he expected "several more days of tough urban fighting."
The U.S. and Iraqi leaders in Iraq have said Fallujah is a base for some 3,000 insurgents, including foreign Islamist extremists and Saddam Hussein loyalists. The insurgents have resisted the occupation of coalition forces and the authority of the U.S.-backed Iraqi government through attacks on military and civilian targets and the kidnapping and execution of foreign civilians.
Abu Musab al-Zarqawi, a Jordanian terrorist thought to be personally responsible for a number of hostage beheadings, has reportedly been operating from Fallujah. Metz said Tuesday that Zarqawi may have escaped the city before the offensive began.
Iraqi Prime Minister Iyad Allawi has said Fallujah must be brought under control to ensure the effectiveness of national elections scheduled for January.
However, some Sunni Muslim clerics, upset over the attack on Fallujah have reportedly urged a boycott of the elections. The elections will be held "over the corpses of those killed in Fallujah," said cleric Harith al-Dhari, according to the Associated Press.
As troops advanced into Fallujah on Tuesday a rising wave of violence in the Iraqi capital of Baghdad prompted Allawi to declare a curfew.
Several Iraqi civilians and U.S. soldiers were killed by insurgent attacks in and around Baghdad on Monday and Tuesday, and a number of bombings were reported throughout Baghdad Tuesday evening local time. Insurgent attacks also were reported in the cities of Ramadi, Hit, Mosul and Haditha.
U.S. military officials have said they expect continued fierce fighting in the battle for control of Fallujah.
"These folks are determined. These are killers. They chop people's heads off," said U.S. Secretary of Defense Donald Rumsfeld.