The Society of Jesus, founded in 1540 by Ignatius of Loyola -- a former Spanish soldier -- is sometimes depicted as an "army of shadows".
Conspiracy theorists say the order controls everything from the Vatican's governing body, to the United Nations and the US banking system.
Rumours of the head of the Society of Jesus's influence in the Holy See have led to the person who occupies the post being nicknamed "black pope".
While some claim the moniker refers to the order's sombre garb, others claim its leader actually pulls the pope's strings.
The society's most famous motto -- Perinde ac cadaver (Obey as a dead body obeys) -- has also fuelled speculation.
The rumours are based on "Jesuit self-discipline, their total submission, their rules on secrecy and their utter obedience to the pope," said Henri Tincq, former Vatican expert for French daily Le Monde.
The society was also feared by some in past centuries for its "desire to influence the bourgeoise elites, who were handpicked and groomed" in special schools, he said.
Italian Catholic historian Alberto Melloni said the society's murky reputation was unfair, insisting "it is a large family where there is a bit of everything -- great conservatives and great reformers."
The order was disbanded by Pope Clement XIV in 1773 amid concern over its members' prowess as political operators and educators of the intellectual elites.
It was restored again in 1814 by Pope Pius VII as a counter to anti-clerical influences in Europe -- but experts say whatever power it now has, it is unlikely to have influenced the election of the new pope, Cardinal Jorge Mario Bergoglio.
"It's not as if it was a club of Jesuit cardinals which elected him. He was the only Jesuit in the conclave," priest Louis Boisset told AFP, laughing off the idea that an order with just 19,000 members dispersed in 150 countries could have such power.
Vatican spokesman Federico Lombardi, who is also a Jesuit, said members of the order "see themselves as subordinates rather than a Church authority," adding that he believed Bergoglio's election was just "a pressing call for him to put himself at the service of the universal Church."
Academic Jose Bento da Silva, an expert on Jesuits at Warwick Business School in Britain, said: "No other religious order knows how to manage a global community like the Jesuits.
"The Jesuits are strongly aligned with the Catholic Church's doctrine and famous for unsettling the hierarchy of the Church," he said.
At the last conclave in 2005, Bergoglio had been a strong frontrunner for the papacy but had bowed out to leave the path free for Benedict XVI to become pope.
This time around, it was he who emerged to the ecstatic crowds in St. Peter's Square as the new leader of the world's 1.2 billion Catholics.
If he failed to appear overly emotional at his election, "that comes from the practice of 'holy indifference'" -- the Jesuit art of accepting that God's will be done rather than your own, Melloni said.
Antonio Spadaro, editor of the Jesuit Catholic Civilization periodical, said Bergoglio's decision to pay homage to St Francis of Assisi -- who was born into a well-off family but chose to live in poverty -- was significant.
Poverty "is at the heart of the Jesuit experience" and the society's founder Loyola followed Francis of Assisi's example when he underwent his spiritual conversion, he said.
Loyola turned to God after being hit by a cannonball and wounded in the leg while defending the citadel of Pamplona against the French in 1521.
He later lamented having been "a man given to the vanities of the world, whose chief delight consisted in martial exercises, with a great and vain desire to win renown."
According to Spadaro, Pope Francis "will have a vision of the world inspired by his Jesuit vocation, but it will be the Society of Jesus which will be in his service from now on."