Iraqi government forces have seized several villages as they move towards an assault on the last area held by the so-called Islamic State (IS) in Mosul.
Hundreds of military vehicles, backed by air power, rolled across the desert towards IS positions early on Sunday.
The progress on Sunday in the south of the city, the second biggest in Iraq, takes them within striking distance of Mosul airport.
Fears have been voiced about the safety of many thousands of trapped civilians.
The offensive was formally announced by Iraqi Prime Minister Haider al-Abadi early on Sunday.
Army Staff Lieutenant General Abdulamir Yarallah said in a statement that elite Rapid Response units captured the villages of Athbah and Al-Lazzagah – two villages south of Mosul airport.
Government forces retook the eastern side of the city, the last major IS stronghold in Iraq, last month. But military officials say the western side, with its narrow, winding streets, may prove a bigger challenge.
For now, there is no advance from eastern Mosul as all bridges from there to the west of the city, across the Tigris river, have been destroyed.
Lieutenant General Stephen Townsend, the commander of the US-led coalition forces, said in a statement on Sunday: “Mosul would be a tough fight for any army in the world.”
Special forces units safely detonated a number of IS car bombs as they cleared villages south of Mosul, according to the BBC’s Quentin Sommerville, who is embedded with the troops.
As well as primed car bombs, the jihadists left behind SIM cards, clothes, instant coffee and weapons as they retreated.
The UN has voiced concern about civilians trapped there, amid reports that they could number up to 650,000. Leaflets warning residents of an imminent offensive were earlier dropped over the west of the city.
Charity Save the Children said on Sunday it believed that as many as 350,000 children were trapped.
“This is the grim choice for children in western Mosul right now: bombs, crossfire and hunger if they stay; or execution and snipers if they try to run,” said the charity’s Iraq country director, Maurizio Crivallero.
The BBC’s Quentin Sommerville is embedded with the Emergency Response Division special forces troops near Mosul
The assault began just after dawn, after days of coalition air strikes, with hundreds of armoured vehicles, thousands of men, and support from helicopter gunships.
The men of Iraq’s Emergency Response Division, police special forces are leading the attack. Their targets are three IS held villages to the south of west Mosul. They are trying to gain the high ground from IS, which will give them sight of the city’s airport and its southern edge.
The government forces made quick gains but have been slowed as they begin to take villages. There are no signs of any civilians. Heavy machine gunfire, rockets and artillery fire are constant.
For the first time in nearly three years, the Iraqi flag is again flying over the south of western Mosul.
Iraqi forces have now all but surrounded the western part of Mosul, while the US-led coalition has been carrying out air strikes on IS targets.
Ahead of the launch of the operation, Mr Abadi said in a televised speech: “We announce the start of a new phase in the operation, we are coming to Nineveh to liberate the western side of Mosul.”
“Our forces are beginning the liberation of the citizens from the terror of Daesh [IS],” he added, quoted by AFP news agency.
The UN humanitarian co-ordinator for Iraq, Lise Grande, told the BBC on Saturday that “all of the parties to the conflict do absolutely everything they can to ensure that civilians survive the battle, and that they live”.
“Absolutely nothing is more important going into the campaign to retake western Mosul,” she added.
As the advance got under way, the UN commissioner for human rights called on the Iraqi government to investigate videos shared on social media that appeared to show Iraqi troops brutally abusing and executing IS fighters on the streets of east Mosul late last year.
The videos have not been verified at this stage by any government authority or independent group. The Iraqi prime minister’s office said it had launched an investigation.
The offensive on the eastern part of the city was launched on 17 October, more than two years after jihadists overran Mosul before seizing control of much of northern and western Iraq.
Experts warn that western Mosul, although slightly smaller than the east, is more densely populated and includes districts that are seen as pro-IS.
The UN said in late January that almost half of all the casualties in Mosul were civilians. At least 1,096 have been killed and 694 injured across Nineveh province since the start of October.