On May 24, 2014, a man opened fire inside the Jewish Museum in Brussels, quickly killing three people and fatally wounding a fourth before disappearing into the city’s streets. The alleged perpetrator, a French citizen named Mehdi Nemmouche, who has since been arrested and charged with murder, had spent the previous year fighting with jihadist opposition groups in Syria. His attack appeared to mark the first time that the Syrian civil war had spilled over into the European Union. Many security officials in Europe and the United States fear that this strike foreshadowed a spate of terrorist attacks that the chaos in Syria—and now Iraq—could trigger.

The Syrian conflict has captured the imaginations and inflamed the passions of Muslims around the world, spurring thousands to join the mostly Sunni rebels resisting the Assad regime. The influx of volunteers has bolstered jihadist groups such as the Islamic State of Iraq and al-Sham (ISIS), also known as the Islamic State, a militant organization that swept across Syria’s border into Iraq this past summer and proclaimed an Islamic caliphate.

Although most foreign fighters in Syria and Iraq come from the Arab world, a sizable contingent hails from the West’s large Muslim communities; 19 million Muslims live in the EU, and more than two million call the United States home. Since the beginning of the Syrian civil war, about 2,500 people from those places (as well as Australia, Canada, and New Zealand) have traveled to Syria to fight, according to the Soufan Group, a U.S. security consulting firm.

Intelligence officials fear that these volunteers might return from the battlefield as terrorists trained to wage jihad against their home countries. Echoing these worries, Charles Farr, the director of the British Office for Security and Counter-Terrorism, described the Syrian war this past summer as “a very profound game changer” for the extremist threat to Europe. Similarly, James Comey, the director of the FBI, warned in May that the repercussions from the conflict might be “an