By William Horsley
European affairs correspondent, BBC News
On the fifth anniversary of the 11 September attacks against the US, Europeans agree with Americans that terrorism inspired by Muslim fundamentalism is a big threat to their lives.
Europe has had to face up to the existence of radical Islam
That new fear, combined with alarm at the conflicts on Europe's doorstep in the Middle East and serious European doubts about US global leadership, means Europe as a whole is marking the anniversary in a mood of pessimism and uncertainty.
That is reflected in the statements of European leaders on the anniversary.
The government of Finland, which now holds the presidency of the 25-nation European Union, condemned all forms of terrorism, saying that "no cause, no grievance, can justify" any terrorist acts.
But Germany's Chancellor Angela Merkel, echoing European anguish over reports of secret CIA jails and alleged torture in Europe, said: "Our battle against Islamic terrorism will only succeed if we cultivate respect for human rights."
Europe's double disillusionment - with its US ally and with the reported growth of fanatically violent Muslim groups in its towns and cities - is apparent from opinion polls.
A survey in the US and 12 European countries, released a few days ago by the German Marshall Fund of the US, found that disapproval of US handling of international affairs among Europeans had reached a new peak of 77%.
Europeans are also much more fearful of Islamic fundamentalism, with 56% now identifying it as an "extremely important" threat (compared with 58% of Americans), and another 34% seeing it as an "important threat" (Americans 31%).
Bombs in Europe
Five years ago, the French newspaper Le Monde coined the phrase "We are all Americans now" to express Europe's overwhelming sympathy with the US after the attacks on New York and Washington.
That emotional bond and sense of solidarity largely evaporated in the years that followed, as European public opinion turned against America's way of conducting the "war on terror" - especially the invasion of Iraq and human rights abuses associated with the Abu Ghraib prison and Guantanamo Bay detention camp.
Europeans had to face the discovery that the 11 September al-Qaeda plot was planned by a group of militant Arab Muslim youths in the German city of Hamburg.
The Madrid bombs were the first al-Qaeda-inspired attacks in Europe
But at first a belief persisted that European countries would not be targeted, provided they did not actively help the US army in Iraq.
European governments began building common defences against acts of terror, including a cross-border European arrest warrant.
But it was not enough. The deaths of 191 people in the Madrid train bombings of 2004 were followed by the killing of 52 innocent people in suicide bombings on London's transport system the next year.
Both outrages were found to be the work of young Muslims imbued with hatred for the West.
The London bombings marked the first case of Islamic suicide bombings in Europe. It also proved the existence of a "home-grown" terrorist threat: four of the bombers were young British Muslims of Pakistani descent.
In a pre-recorded video, one called himself a soldier who wanted to avenge "my Muslim brothers and sisters" in places like Iraq and Afghanistan.
Security experts believe Osama Bin Laden's al-Qaeda network has inspired an unknown number of "self-starter" cells, many of them in Europe, each of which has a host of potential targets for attack.
Recent threats include an announcement by German authorities of what they called evidence of the gravest threat so far: self-made suitcase bombs on passenger trains.
Bombings in London put a strain on the UK's Muslim community
Danish police said they had seized chemicals that could be used to make bombs during the arrest of a group of young Muslim men suspected of planning a terrorist act.
And British security services exposed an alleged plot to blow up transatlantic passenger planes flying out of London's Heathrow Airport, with a loss of life which officials said could have been greater than in the attacks on New York's World Trade Center.
The UK's top anti-terrorism officer says the number of those suspected of actively supporting terrorism is "in the thousands".
In Europe, no country now thinks of itself as immune.
In an interview to mark this anniversary, France's leading anti-terrorism judge, Jean-Louis Bruguiere, said France was "indisputably" among the possible targets and the threat was still at a high level.
The most recent evidence of the spread of what commentators have called a "cult of death" among alienated young Muslims in parts of Europe has sharpened the debate.
A wide gulf still divides mainstream opinion among the non-Muslim majority in Europe and most of Europe's 15 million or more Muslims
European governments, acting by themselves or together through the European Union, are taking steps to ensure that mosques are not used as places to foster political violence or to recruit people to extremist causes.
The search is on for a "European Islam", untainted by political fundamentalism.
The British media, like others around Europe, recently printed many articles harshly condemning the ideology now being spread in the name of Islam.
David Selbourne, author of a book called The Losing Battle with Islam, wrote of Islam's "moral intransigence, its jihadist ethic and the refusal of most diaspora Muslims to 'share a common set of values' with non-Muslims".
And Dr Maha Azzam of Chatham House, a leading London-based think-tank, says al-Qaeda is facing a "very serious challenge to its legitimacy".
Because of its terrorist activities, Dr Azzam writes, al-Qaeda has also lost popularity in the Muslim world.
Yet a wide gulf still divides mainstream opinion among the non-Muslim majority in Europe and most of Europe's 15 million or more Muslims.
Muslims in Europe believe Islamophobia is on the rise.
The poll by the German Marshall Fund of the US found as many as 56% of all Europeans now see Islam as "not compatible with their democracy".
On that point, too, Europeans and Americans, for all their differences over foreign policy, now see things the same way.