The strategy of the US and the Iraqi interim government to retake the rebel Sunni Muslim areas is clear enough in concept - but will be hard to carry out in practice.
By Paul Reynolds
BBC News Online world affairs correspondent
The plan is to re-establish government control in advance of the elections planned for January.
US troops are trying to regain control of Samarra
Samarra has been the first of a series of targets. The bombing in Falluja, regarded as the hardest nut to crack, is reckoned to be a prelude to a similar ground assault there, probably after the US elections in November.
The problem is that a military assault is one thing and establishing effective civilian government is another.
But while it is not an easy job, it is, for the US and the Iraqi interim government, an essential one.
If control is not established, the holding of elections in these places will be in doubt. And if large numbers of Sunni Muslims cannot vote, there is the risk of them becoming further alienated after the January poll.
If that happens, the task of integrating Iraq's three constituencies of Sunnis, Shias and Kurds will be even harder to achieve.
"The election is for the US a defining moment," said Toby Dodge, Iraq watcher at Queen Mary College, University of London. "It is supposed to deliver the next stage after the June handover and some form of valid control over these places is needed.
"But the presence of the government in Iraq is a fleeting thing," he said. "The state is not strong or coherent enough to go in and stay. The US military can hit towns but there are no civil institutions to run them.
"Samarra was the least difficult of the rebel cities. It has a history of strong opposition to Saddam Hussein and could be called the easy first stage. The US military has been in and out before. It has now gone in to seize it. The bigger question is how you hold it.
"Falluja is also being softened up in readiness for an attack. But there are a lot of angry people there and it is a big town. It will take a lot of troops and will be a violent struggle."
Eyewitness in Falluja
The situation inside Falluja is difficult to judge, given that few reporters get there.
One very revealing account has come from the Institute for War and Peace Reporting, a media monitoring and action group based in London. It shows the extent of the task ahead.
According to one of its Baghdad staff, Dhiya Rassan, Falluja is controlled by a number of differing groups from "foreign fundamentalists to former Saddam loyalists".
These include "the most-feared insurgents" called the Black Banners Brigades of the Islamic Army, led by Omar al-Hadid, said to have links to al-Qaeda.
"The few police on the streets of the towns are entirely at the insurgents' beck and call," Dhiya Rassan reported.
"Citizens of the town live under the often capricious rule of different groups of mujahideen, or holy warriors - ranging from Islamists and ultra-Islamists to Baathists and outright bandits.
"Nonetheless, residents say the groups are united on the battlefield and would fight side-by-side if US or Iraqi government troops were to launch a new push."
The building of a political system there is a obviously huge challenge.
Absence of politics
But even in areas under government control, the development of politics in advance of the elections has been slow.
The political parties formed by exiles returning home, sometimes after many years abroad, have not attracted much support.
There is now some talk of new alliances being contemplated or formed.
For example, among the Shias, the largest population group, there is manoeuvring around the maverick figure of Moqtada Sadr, the young cleric who led an uprising earlier this year.
He has hinted he might enter the elections, and other Shia parties are said to be courting him on the condition that he disband his militia.
In the meantime, figures compiled by Washington think-tank the Brookings Institution as part of its invaluable Iraq Index - which tracks a number of key indicators - show how insurgent attacks against US and other foreign forces have increased this year.
At the start of 2004, they were actually going down, from 735 last November to 410 in February this year. Since then there has been an increase to 2,700 in August and 2,400 in September.
Such figures can only give a rough outline of what is going on. Some of the worst attacks are against Iraqi civilians.
But they do indicate the nature of the problems being faced at a time when it had once been hoped that politics would be emerging as violence was disappearing.