ISIS, as the Islamic State jihadists in Iraq and Syria are known, has become the new face of international terrorism in the eyes of the United States and its Western allies.
Now the focus in America and abroad has become what will President Barack Obama and other leaders do about it?
Here are key questions on the matter:
1. Who assassinated James Foley?
Britain's ambassador to the United States, Peter Westmacott, told CNN on Sunday that British officials were close to identifying the ISIS militant who beheaded Foley, an American journalist captured in Syria in 2012.
He couldn't elaborate on the identity of the killer, who is seen decapitating Foley in a video posted last week on YouTube.
"We're putting a great deal into the search," he said, referring to the use of sophisticated technology to analyze the man's voice.
In the video, Foley, 40, is seen kneeling next to a man dressed in black, who speaks with what experts say is a distinctly English accent.
Linguists said that based on his voice, the man sounds to be younger than 30. He also appears to have been educated in England from a young age and to be from southern England or London.
2. Will the United States expand air strikes to ISIS targets in Syria?
Defense Secretary Chuck Hagel and Joint Chiefs Chairman Gen. Martin Dempsey both said last week that taking on ISIS in Syria was the only way to defeat the Sunni jihadists.
On Sunday, Republicans voiced a chorus of calls for the United States to do exactly that.
For Obama, the step would reverse his refusal for three years to get involved militarily in Syria despite pressure from his own advisers, including former Secretary of State Hillary Clinton.
CNN National Security Analyst Peter Bergen said it will be difficult to defeat ISIS without ground forces, something Obama clearly opposes. Intervening in Syria also could result in some strange geopolitical bedfellows, he noted.
"Two of the most effective fighting forces in Syria are al Qaeda or al Qaeda splinter groups, or groups like Hezbollah, backed by Iran," Bergen said. "So if you intervene, you may be helping Iran and Hezbollah and (Syrian President Bashar al-Assad's) regime."
Obama already sent military advisers to Iraq and launched air strikes to protect them and minority groups from ISIS fighters rampaging through the country's north.
Pentagon officials say they are considering options to expand the air strikes into Syria, but no decision has been made.
A White House spokesman said last week that Obama would consult with Congress before taking such a step. The President also would seek to forge a coalition including regional allies as well as U.N. and European Union support, officials have made clear.
3. Will the Syrian regime that Obama opposes help fight ISIS?
Obama wants al-Assad out of power, but now the Syrian leader engaged in a civil war against a U.S.-backed opposition is offering to help him take on ISIS.
Foreign Minister Walid Moallem said Monday his government would accept support from the United States and others working under the U.N. umbrella to fight "terrorists" -- a code word for the group that calls itself the Islamic State and seeks to establish a caliphate across a Sunni-dominated swath of the the Middle East.
Moallem, however, warned against any unilateral action or strikes in Syrian territory without its permission, saying "any effort to fight terrorism should be done in coordination" with the "Syrian government."
Last week, Atlantic Council senior fellow and Syria expert Frederic Hof said a U.S. rescue mission for Foley earlier this year that went into Syria but failed to find him established the precedent for military action across the Iraq border, superseding any legal considerations such as being asked by the host government to enter.
"The sort of legal barrier that prohibited doing something inside Syria now seems to have evaporated," Hof said.
The Syrian offer to help fight ISIS comes after al-Assad's government enabled the group to expand amid the Syrian civil war. ISIS fighters have attacked the Syrian opposition fighting government forces, but also have seized some government territory.
Al-Assad's military recently launched its own air strikes on ISIS positions, amounting to what Hof described as a dispute between crime gangs over money -- in this case, from oil fields occupied by ISIS.
4. Will ISIS attack the West?
To some in the United States, especially critics of Obama, an ISIS attack on U.S. interests and even the homeland is a question of when, not if.
"ISIS is a very powerful local organization, and probably a reasonably powerful regional terrorist organization," former CIA
chief Michael Hayden told CNN on Sunday. "But it's one that has global ambitions -- and it has the tools."
There's no clear consensus inside the intelligence community as to whether ISIS, which calls itself the Islamic State, is currently capable of striking the West.
"It's expressed the intent," Hayden said. "There's no more powerful way to express their street credentials among the jihadist community than a successful attack against the West."
Sen. Lindsey Graham, a hawkish Republican from South Carolina, told CNN on Sunday that "it's about time now to assume the worst about these guys, rather than to be underestimating them."
5. Can the ISIS money flow be stopped?
Bank robbery, kidnapping, smuggling, selling oil on the black market -- ISIS gets money to fund and expand its organization in all kinds of ways.
Officials say the group can get about $3 million a day by selling discounted oil from fields it has seized in Iraq. It also has grabbed millions robbing banks including an Iraqi central bank in Mosul.
Western allies can reduce the group's income by refusing to pay ransom for abducted citizens and pressuring regional governments to crack down on wealthy citizens sending money to it.
The United States is working with governments in the region, including Kuwait, Qatar and Saudi Arabia, to stop such private donations, State Department spokeswoman Marie Harf said last week.