The group's death squads have stalked Turkey since the 1960s, murdering left-wing and liberal activists, university students and journalists and engaging in street battles and attacks.
They gained international notoriety in 1981 when Mehmet Ali Agca, one of their collaborators, shot and nearly killed Pope John Paul II in St Peter's Square.
The Grey Wolves are known for their distinct hand sign, which represents a wolf head, made by holding up a forefinger and little finger.
The group's ideology centres on the glory days of Turkish history, seeking to unite Muslim Turkic peoples from the Balkans to Central Asia in a pan-Turkish extension of the Turkish nation-state.
The group extended operations in the early 1990s into post-Soviet states with Turkic and Muslim populations, including the Nagorno-Karabakh war in Azerbaijan and Chechen conflicts.
The group is believed to have ties to Turkish crime gangs that operate in Bangkok that could have provided logistical support for their attack, security analysts say.
Thai police have been searching for Turkish nationals who arrived in Thailand in the 15 days before a blast tore through foreign tourists and Thais at the Erawan shrine on August 17, killing 20 people and injuring more than 120 others in an unprecedented attack.
But their breakthrough in the investigation came when residents of a predominantly Muslim district of Bangkok on Saturday reported to police the suspicious activities of a non-Thai speaking man renting five rooms in a seedy, four-storey apartment block.
After more than 100 police surrounded the building they found a man believed to 28 years old in a room with a stack of false passports and bomb making equipment similar to that used in the shrine bombing, including ball bearings, pipes and fuses.
The bearded man with short cropped hair has been charged with possession of bomb making material and is being held in a Thai military base pending further investigation.
Anthony Davis, a respected Bangkok-based security analyst with IHS-Jane's, said last week the Grey Wolves were likely to be behind the bombing because they had both motive and capability, although he did not rule out other possibilities.
"They are violent and operate below the radar," he said.
Mr Davis said the group had "latched on to in a big way" Uighur Muslims in western China who claim they have suffered years of persecution from Beijing.
Thailand infuriated the Uighur movement in July when the country deported to China 109 Uighur men who had been separated from their wives and children.
Ethnic-Chinese tourists appear to have been targets of the shrine bombers.
Mr Davis described the Bangkok attack as potentially the nightmare that has worried security agencies, a link-up between terrorism and organised crime.