They were to be found, Bush said on 15 September 2001, in the caves and mountains of Afghanistan.
"We will find those who did it [the terror attacks in New York and Washington]," Bush said. "We will smoke them out of their holes. We will get them running. And we will bring them justice."
Four years ago this week -- on 7 October 2001 -- the United States launched air strikes on Taliban and Al-Qaeda positions to begin the U.S.-led invasion of Afghanistan. Washington's goal was to topple the Taliban, destroy terrorist bases, and capture or kill Al-Qaeda leader Osama bin Laden.
Except for bin Laden, whose fate remains unclear, all those objectives were achieved.
And yet, in the last four years, attacks have multiplied. Terrorists have struck in Morocco, Turkey, Russia, Spain, Uzbekistan, and Indonesia -- the latest one killing 26 people two weeks ago in the resort of Bali.
On 7 July this year, coordinated suicide strikes on the London metro killed 52 people. And last week in Paris, police announced the arrest of nine people suspected of plotting similar attacks on the French capital's subway system.
So if Al-Qaeda's central command was seriously damaged after 9/11, what is behind the growing global wave of terrorism by extremist Islamist groups?
Jerrold Post directs the political psychology program at George Washington University in the U.S. capital. Post, who spent two decades with the CIA, told RFE/RL that after the Afghan war, bin Laden sent a message around the world asking sympathizers to "go forth and multiply." Post said the result is a growing global "Salafi jihad" run not so much by a central organization as it is inspired by a common ideology.
"It really has changed into Al-Qaeda 2.0 [ed.: a new version], which some would say is mainly an ideology; others would put it more organizationally," Post said. "But it's more a network, loosely connected and rather semiautonomous."
Post said there's a "tension" or debate among experts as to what extent cells such as the London bombers are directed by a central command or carried out independently.
General John Abizaid, head of U.S. Central Command, painted a similar picture last month during testimony before the U.S. Senate Armed Services Committee.
"It [Al-Qaeda] is not like IBM, a monolith that is centrally led from a central headquarters," Abizaid explained. "It is much more like McDonalds, a franchise that is decentralized and dangerous and linked in many ways."
But Abizaid said Al-Qaeda continues to carry out key functions, even as terrorist cells go about business on their own.
"They've developed a media and a propaganda campaign...and Internet and proselytizing campaign, recruitment and education; they develop safe havens that are both geographic and ungoverned spaces and virtual within the Internet and within the mass media world," Abizaid said. "They have front companies. They buy off politicians. They develop facilitators and smugglers. They deal with financiers that move drug money around as well as other illicit money. And they have sympathetic, nongovernmental organizations that they sponsor to transfer some of their hateful ideology in very, very insidious ways around the region."
Still, "they" appear to be not so much a single entity as a constellation of like-minded groups such as Jemaah Islamiyah. The Indonesian organization is a suspect in the recent Bali blasts and claims links to Al-Qaeda.
Then there's Hizb-ut Tahrir. It shares goals with Al-Qaeda, such as the establishment of an Islamic caliphate throughout the Muslim world. But the group says it rejects all forms of violence, even as Germany and Britain have moved to ban it for allegedly inciting terror.
Uzbek officials frequently rail against Hizb-ut Tahrir, which enjoys a growing number of followers in Central Asia, and blame it for several attacks -- all duly denied by the group. Uzbek Foreign Minister Elyor Ganiev appeared to have Hizb-ut-Tahrir in mind when he addressed the United Nations General Assembly on 16 September.
"We are convinced that terrorism cannot be defeated only by eliminating and neutralizing separate terrorist groups or terrorists," Ganiev said. "It is obvious that notable results can be achieved only if ideological centers which nourish, finance, and organize forces of international terrorism are eradicated."
But wiping out "ideological centers" is a tall task in the Internet age. Magnus Ranstorp, an expert on terrorism and political violence at Saint Andrews University in Scotland, told RFE/RL that the web, never easy to police, is the communication medium of choice for militants around the world.
"Cyberspace is a fantastic, passive medium to project your ideology, to project your messages in an uncensored format," Ranstorp said. "It is a fantastic medium to really sort of perform psychological warfare against the West -- but at the same time recruit, mobilize the community."
In the days after the 7 July attacks in London, many analysts said the strikes were an example of how decentralized terrorism by extremist Islamist groups has become.
But in early September, a video surfaced featuring the alleged ringleader of those attacks as well as bin Laden's deputy, Ayman al-Zawahri, who claimed Al-Qaeda responsibility for the London bombings.
Michael Scheuer, the former head of the "bin Laden unit" at the CIA, said this week that he believes al-Zawahri has now taken a high-profile role in Al-Qaeda because bin Laden is waiting to resurface.
Bin Laden is still believed to be hiding somewhere between Afghanistan and Pakistan, his communication network shattered.
But Scheuer, in an interview with AFP in Washington, said Al-Qaeda remains powerful. He said he believes bin Laden is biding his time out of the public eye -- but plans to return after another spectacular attack on U.S. soil.
The third part of our series looks at how countries waging the war on terror balance their need for security with their desire to maintain civil liberties.
(RFE/RL's Uzbek Service contributed to this report)
World: Four Years Later, No Clear Winners In War On Terror (Part 1)
World: Who Is The Enemy -- And How Is It Changing? (Part 2)
World: Can West Fight Terror And Still Maintain Civil Liberties? (Part 3)
World: Global War On Terror Faces Government With Unique Enemy (Part 4)
For all of RFE/RL's coverage on the global fight against terrorism, visit our War On Terror website.