President Obama refuses to call the group that has overrun vast tracts of Iraq and Syria by their chosen name - so how should we refer to them?
The militants who have declared a Muslim "caliphate" in parts of Syria and Iraq call themselves the Islamic State - but detractors say they represent neither Islam nor any state.
In a speech outlining his strategy against the group on Wednesday, President Barack Obama pointedly denied the extremists their chosen appellation.
He has for weeks been referring to them by the acronym ISIL, which stands for the Islamic State of Iraq and the Levant, and is a translation of a name IS used before declaring their caliphate in June.
"ISIL... calls itself the 'Islamic State'", Obama said.
"Now let's make two things clear: ISIL is not 'Islamic.' No religion condones the killing of innocents. And the vast majority of ISIL's victims have been Muslim," he added.
"And ISIL is certainly not a state. It was formerly al-Qaeda's affiliate in Iraq, and has taken advantage of sectarian strife and Syria's civil war to gain territory on both sides of the Iraq-Syrian border."
That predecessor group was known as the Islamic State of Iraq, and was formed in October 2006, when the group known as al-Qaeda in Mesopotamia merged with various Iraqi Islamist groups.
After years of vicious battle in Iraq, the group's influence waned after many of Iraq’s Sunni tribes turned against them.
But the civil war in neighbouring Syria offered the remnants of the organisation an opportunity to re-form in another guise and under a new name.
As early as August 2011, the group was calling on its backers to go to Syria to fight alongside "the Muslims" against Alawites, a sect with roots in Shiite Islam to which Syria's President Bashar al-Assad belongs.
The group dispatched members from Iraq to Syria to help found al-Nusra Front and in April 2013, the leader of the Islamic State of Iraq announced that the groups would merge to form the Islamic State of Iraq and the Levant.
But al-Nusra's leader rejected the merger, as did al-Qaeda chief Ayman al-Zawahiri, who designated al-Nusra as al-Qaeda's official Syria affiliate, ordering the Islamic State of Iraq to return across the border.
The order was ignored, and the new name came to refer to a militant group separate from al-Nusra, with cross-border aspirations and capacity.
The naming created confusion for some.
The Arabic word for Levant, al-Sham, can also be used to indicate Syria alone.
So some people chose to call the group ISIL, standing for Islamic State in Iraq and the Levant, while others opted for ISIS, Islamic State in Iraq and Syria.
ISIL or ISIS - the great debate
In the United States, the Washington Post reported that Democratic lawmakers have officially decided to use ISIL because Isis is the name of an ancient Egyptian goddess as well as a female name.
Republicans, however, appear to favour ISIS in their references to the group, according to the newspaper.
In Arabic, the group has come to be known by many as Daesh, an acronym for the group's full Arabic name.
The group’s fighters consider the term derogatory, because it removes the "Islamic" part of their title.
Both French President Francois Hollande and French Foreign Minister Laurent Fabius have publicly used "Daesh" when referring to the group.
And in some parts of the Arab world, including countries as far apart as Lebanon and Yemen, it is now used as an adjective - a singular "daeshi" or plural “dawaish”, in reference to a bigot who imposes their views on others.
Last month, Saudi Arabia's highest religious authority, Grand Mufti Sheikh Abdul Aziz al-Sheikh, branded IS militants and al-Qaeda as "enemy number one" of Islam.
And Egypt's al-Azhar, the top authority in Sunni Islam, has urged foreign media to stop referring to the militants as the Islamic State.