By Nick Childs
BBC world affairs correspondent
Reports and rumours continue to circulate about the condition and whereabouts of the Jordanian-born insurgent leader in Iraq, Abu Musab al-Zarqawi.
Zarqawi (centre) is the Americans' top target in Iraq
It remains unclear whether he is seriously wounded, whether he is in Iraq or elsewhere, and whether he has been replaced at least temporarily as leader of his group.
But what is his significance and the potential impact of his fate on the insurgency?
The US military in Iraq makes no secret of the fact that Zarqawi is its number one target.
He is blamed for many of the most deadly attacks and is the only widely-recognised leader in the insurgency.
Clearly, US commanders and the new Iraqi authorities believe that removing him from the scene would have a significant psychological and practical impact on the insurgency, at least temporarily.
The Saudi authorities have announced successes against certain leaders, only for others to come forward
To capture or kill such a figurehead would also be a considerable symbolic and morale boost to US forces and the Iraqi administration.
But no-one believes it would be a silver bullet that would finish off the resistance.
US officials acknowledge that they and the Iraqi authorities will not defeat the insurgency by military means alone.
It also requires a broader strategy involving politics and economics. And it will probably take years.
The US military claims to be cracking the insurgency
In that context, there is growing concern about the sectarian fallout of the violence, and a particular focus now on trying to damp down these tensions, and in particular engage the Sunni community in the political process.
On top of that, the experience of the Saudi authorities in dealing with militants in their country has been that they have announced successes against certain leaders, only for others to come forward.
In Iraq, the insurgency remains a complicated tapestry. As it has gone on, it has got more sophisticated.
The foreign fighters whom Abu Musab al-Zarqawi leads may be only a portion of the total.
But trends like the increase in suicide bombings in Baghdad may be evidence of growing collaboration between the foreign elements and local Iraqi insurgents.
US tactics have included continuing offensives against insurgent strongholds, to try to keep them off balance.
The recent operations - codenamed Matador and New Market - in the west of the country, and the joint US-Iraqi mission in the western suburbs of Baghdad, dubbed Squeeze Play, are examples of that.
Have US planners underestimated the strength of their opponents?
They have targeted and captured senior Zarqawi associates, they say, and believe they may even have got close to the man himself.
And yet the level of insurgent violence has gone up.
That may be, as the Americans contend, a sign of increasing desperation.
Or it may be further evidence that the insurgency is stronger and more sophisticated than US commanders and intelligence have calculated.