The recent defeat of Islamic State (IS) forces in the Syrian border town of Kobane has been greeted by the US-led coalition fighting the group as a significant victory.
But the killings of two Japanese civilians and a Jordanian pilot held prisoner by IS suggest it shows no sign of backing down in the face of five months of American-led airstrikes in Syria and Iraq.
So how much progress has the military offensive against IS made?
The defeat of IS at Kobane was achieved by a combination of Syrian and Iraqi Kurdish fighters on the ground supported by more than 700 air strikes which have flattened much of the town.
This success supports the view of many analysts that to defeat Islamic State, air strikes alone are not enough and ground forces are essential if the organisation’s territorial gains are to be rolled back.
The size of the challenge this presents is made stark by the assessment of informed observers that contrary to what President Obama told Congress in his State of the Union speech last month, since the campaign against IS began last August, the Islamist fighters have in fact doubled the amount of territory they control, especially in eastern Syria.
IS has also succeeded in gained allies beyond Syria and Iraq. In the past few weeks, Islamists in Egypt and Libya, who have affiliated themselves to Islamic State, have carried out large scale attacks. The group is also reported to be establishing links with the Pakistani Taliban.
Following the taking of Kobane, the US-led coalition, which includes the air forces of several western and Arab states, as well as the Iraqi army and Kurdish Peshmerga fighters, is turning its sights on the Iraqi city of Mosul.
It was the fall of Mosul to IS last June that galvanised Washington to intervene again in the region following the withdrawal of American forces from Iraq at the end of 2011.
If Mosul can be recaptured, it would represent a much bigger blow to IS than Kobane, which it had never fully controlled. But Mosul is a much tougher proposition.
IS has consolidated its control of the city – Iraq’s second largest – and the remaining population is largely Sunni Arab, many of whom may prefer the rule, however harsh, of IS to the return of rule by the Shia-dominated government in Baghdad, despite the efforts of the new Iraqi Prime Minister, Haidar al Abadi, to reach out to disaffected Shias.
It is also unclear whether the Iraqi army, which collapsed in the face of the IS offensive last summer, is yet in a fit state to launch a large-scale ground offensive against Mosul.
The Americans and other western countries, including Britain, have been training and rearming Iraqi forces, but the corruption that is blamed for their cave-in to IS will take time to root out, if indeed it can be.
The lack of an ally with powerful enough ground forces is even more acute in Syria.
Since the uprising against President Assad began four years ago, IS and the al Qaeda-associated al Nusra Front, have come to dominate the rebel forces and American efforts to train rebel bands fighting with the Free Syrian Army, which the US regard as moderate, have been beset with difficulties.
Many rebel commanders have been deemed unreliable by the CIA and have reportedly had their training terminated. Some rebels have returned to Syria only to defect to the Islamist forces which are better financed and armed.
The coalition’s most powerful potential ally in Syria is of course the government of President Assad, and some prominent voices, including Ryan Crocker, former US Ambassador to Syria, Iraq and Afghanistan, have long called on Washington to re-engage with Damascus.
In Iraq, the US is tacitly allied with its long time enemy Iran, which has also sent arms, trainers and special forces and launched air strikes to support the fight against IS. But working openly with the Syrian armed forces would be much more fraught with political difficulty.
The US, its western and Arab allies, have long been saying Mr Assad must stand down, so they would have to eat a lot of very public humble pie if they were to work openly with him now.
The Syrian armed forces have also been accused by western states and human rights organisations of committing war crimes, including the indiscriminate bombing of civilian areas.
There has been speculation that there is a tacit alliance in Syria between President Assad’s forces and the anti-IS coalition, but just to show how complex the situation is, there have also been allegations, especially in Sunni Arab capitals, that Mr Assad has concentrated his forces against secular, so-called moderate, rebels rather than his Islamist opponents, in order to present himself to the world as a bulwark against extremism.
Mr Obama’s critics say the reliance on air strikes and refusal to send American combat troops into action shows the president is half-hearted.
The air campaign, known as Operation Inherent Resolve, has now lasted longer than the operation to topple the Taliban in Kabul following the September 11th attacks and the air strikes against Serbia in 1999 that forced its withdrawal from Kosovo.
But there is another example from the Balkan wars of the 1990s supporters of the current campaign could point to as evidence that a strategy of air strikes plus training and arming allies to fight a ground war can work – assuming you can find a reliable partner.
That is the successful Croatian army campaign in 1995 which destroyed the Croatian Serb statelet of Krajina, drove on into Bosnia and, in combination with NATO air strikes, brought the Serbs to the negotiating table at Dayton and ended the Bosnian war.
In the year before the Croatian offensive, its armed forces were re-organised and trained by former American military personnel working for a private security subcontractor. It is thought a proxy was used to get round a UN arms embargo on the former Yugoslav states that was still in effect. When that embargo was lifted in late 1994, US support for Zagreb became more open and arms supplies flowed in.
In Iraq and Syria, the search for such allies has yielded mixed results so far.
At Kobane, Syrian and Iraqi Kurds have shown they can succeed on Kurdish territory, but to roll IS back, the most eligible ground forces, the Iraqi army, will need to be reformed and reorganised, a task which could well take many more months, if not years.