BBC Middle East analyst Roger Hardy investigates Wahhabism, Saudi Arabia's austere brand of Islam, which is accused by its critics of funding religious extremism.
In the second of his two-part series, he meets top US investigators who have had the frustrating task of finding out where Saudi cash ended up.
In 2002, just a year after the 11 September attacks, the US authorities shut down a Saudi charity headquartered in Chicago.
US investigators believe Saudi donations have funded jihad
"It was a big hoopla," recalls Sam Roe, an investigative journalist who covered the story for the Chicago Tribune.
"The US attorney general at the time, John Ashcroft, flew out to Chicago and held a press conference.
"It was trumpeted as one of the first major victories in the 'war on terrorism'."
The charity was the Benevolence Foundation.
The Benevolence Foundation had been set up in Chicago a decade earlier by a wealthy Saudi businessman, Adel Batterjee.
But he soon handed over the running of it to his right-hand man, a Syrian called Enaam Arnaout.
The two men had met in Afghanistan in the 1980s when both had been helping Muslims fight the Soviet occupation forces.
The charity began to arouse the suspicions of the US authorities in the 1990s. But it was only after 11 September that the case acquired any real urgency - and attracted the attention of a star US attorney, Patrick Fitzgerald.
Speaking to me at his office in an imposing federal building in Chicago, he told me how he became convinced the Benevolence Foundation was an al-Qaeda front.
Documents found at its office in Bosnia, he says, described al-Qaeda's founding meeting, its membership and the shipment of weapons.
But however strong his suspicions, Patrick Fitzgerald could not make the terrorism charges stick.
Wealthy and respected
On the eve of the trial, he struck a deal with Mr Arnaout, who was convicted on a racketeering charge.
He was found guilty of supplying boots and uniforms to Bosnian Muslim fighters, while pretending he was only helping civilians.
Patrick Fitzgerald tried, but could not make terrorism charges stick
It was, at best, an ambiguous victory for the US authorities. They had shut down a charity but had failed to prove its links to al-Qaeda.
While Arnaout serves out his jail term, his former boss Adel Batterjee remains in the Saudi city of Jeddah, a wealthy and respected businessman.
Did he break his ties with Benevolence in the 1990s, as he has always claimed? Or did he run it from afar, as the court documents suggest?
After repeated efforts to hear his side of the story, I spoke briefly to him on his mobile phone - but he declined to be interviewed.
For Saudi Arabia, however, the charity issue refused to go away.
In 2004, at a joint press conference in Washington, the US and Saudi governments announced the closure of five branches of a prominent Saudi charity, al-Haramain.
Eventually 10 branches - stretching from the Netherlands to Indonesia - were designated "financiers of terrorism" and shut down.
Have the Saudis bankrolled the global jihad? They certainly have a case to answer
Set up in the 1990s, al-Haramain was a large and prestigious organisation with close ties to the Saudi government and ruling family.
At its height, it had some 50 branches worldwide.
Its literature proclaimed it had built hundreds of mosques, run orphanages and helped equip clinics and hospitals.
At a tent in a Riyadh suburb, I met the Saudi who had set up a branch of al-Haramain in Ashland, Oregon.
He is now a wanted man.
Sitting on cushions drinking tea, in the company of two American lawyers, Soliman al-Buthi laughed off the charge that he was a "specially designated global terrorist".
"I never met Osama Bin Laden," he told me.
"I have never flown to any of these hot places - Bosnia or Chechnya or Afghanistan or Pakistan."
Adel Jubeir admits the charity commission has not materialised
He insists that he was engaged in the non-violent propagation of Islam - and that no real evidence has ever been presented against al-Haramain.
But for investigators like Dennis Lormel, a senior FBI official who worked on the case, the evidence of links to al-Qaeda was compelling.
Persuading the Saudi authorities to take it seriously, however, was another matter.
It took more than two years of sustained US pressure for them to act.
Eventually 10 branches of al-Haramain were shut down, and its director in Riyadh was sacked.
The Saudi ambassador to Washington, Adel Jubeir, told me the kingdom was not responsible for the overseas branches of its charities.
He acknowledged that a Saudi charity commission responsible for activities overseas - whose birth he himself had announced in 2004 - had still not materialised.
But he insisted that over the last few years Saudi Arabia had taken draconian steps to regulate its charities.
The two case studies show how hard it has been for the American authorities to turn suspicion into proof.
It is a striking fact that, in these and other Muslim charity cases in the US, they have not managed to secure a single conviction on a terrorism charge.
But what's also clear is that, for decades, Saudi charities were virtually unregulated.
Have the Saudis bankrolled the global jihad? They certainly have a case to answer.
The export of Wahhabism - which got going with the oil boom of the 1970s - acquired a dynamic of its own and ended up empowering radical Islamists.
After 11 September, the chickens came home to roost.
The second part of Roger Hardy's series Jihad & the Petrodollar will be broadcast on the BBC World Service on Friday, 23 November.