Osama Bin Laden is the chief suspect, accused of masterminding the 11 September attacks on New York and Washington.
Even before the attacks he was America's most wanted man. But he also commands a large following among many young anti-American Muslims.
He was born in Saudi Arabia in 1956 to a wealthy Yemeni father and Syrian mother, although he was subsequently stripped of his Saudi citizenship.
He founded the network al-Qaeda in 1979, originally as a guesthouse in Peshawar for Arab fighters wanting to join the Mujahideen in opposing the Soviet invasion of Afghanistan. Eventually their numbers became so large that he built new camps inside Afghanistan. The resistance movement was supported by US dollars.
After being expelled from Saudi in 1991 for anti-government activities, he lived in Sudan for five years until he was expelled following US pressure. He went into exile in Afghanistan.
Despite an extensive military operation in Afghanistan it is still not known where he is or even if he is still alive.
Washington has said there is "no question" that he is responsible for the 11 September attacks.
He has also been indicted for the 1998 US embassy bombings in East Africa and the attacks on the USS Cole in October 2000.
The Taleban were accused of harbouring and sponsoring Bin Laden and his associates, who in turn were said to be propping up the Taleban regime before the military offensive swept them from power at the end of 2001.
However, President Bush says his war on terrorism is not just directed against Bin Laden and his al-Qaeda network, but is a broader war against all terrorist groups.
No concrete evidence against Bin Laden has been released by the FBI, CIA or the White House.
But the UK Government has released a document saying intelligence agencies learnt after 11 September that Bin Laden had indicated that he was about to launch a major attack on the US.