Delegates from the largest and some say the most influential Catholic religious order are meeting in Rome this week to begin choosing their new leader, often called the 'black pope'.
The head of the Jesuits - who wears black as opposed to the white of popes - is elected for life and is a highly prestigious figure in the Church.
He commands a 468-year-old organisation which is renowned for its missionary work in tough, frontier locations and its position at the forefront of Catholic culture and education.
Members of the 'Society of Jesus' swear total allegiance to the pope. According to their founder, St Ignatius Loyola, they must show as little resistance to papal orders as a dead body (''perinde ac cadaver'').
Despite its immense prestige, the order has sometimes had a troubled relationship with the Vatican and the term 'Jesuitical' has acquired negative connotations inside and outside the Church.
Some Jesuits have been accused of being overly liberal and of deviating from official Church teaching. Two Jesuit theologians have been rapped by Vatican for their writings in recent years.
In Latin America, left-leaning Jesuits played a role in the development of Liberation Theology, a movement which has been highly controversial in the Catholic Church and which was condemned by Pope John Paul II.
Father Peter Hans Kolvenbach, the 79-year-old Dutchman who has headed the Jesuits for 25 years, is credited with smoothing out the relations with the Vatican after the tensions that characterised his predecessor's dealings with John Paul.
But there are allegedly still fears among conservatives that the Jesuits risk becoming an autonomous branch of the Catholic Church and the order's choice of a new leader will be scrutinised carefully.
When Cardinal Franc Rode', head of the Vatican's congregation for religious institutes, spoke to Jesuit delegates at a Mass to open their conference on Monday, he appeared to voice a note of warning.
''I see with sadness and worry that even among some members of religious families the feeling of being part of the Church is noticeably waning,'' he said.
'VIGILATE OVER DOCTRINE'.
He warned that the growing participation of lay people in Jesuit activities should not ''obscure your identity'' and he called on the order to ''vigilate over the doctrine of your magazines and publications''.
In 2005, the editor of an influential American Jesuit weekly magazine was forced to resign after certain articles irked the Vatican doctrinal watchdog, headed in the preceding years by Joseph Ratzinger, the current pope.
Some 222 delegates representing nearly 20,000 Jesuits around the world have gathered to find a replacement for Father Kolvenbach, a Dutchman, who is stepping down for reasons of age.
A short-list of five or six names has reportedly been prepared but, according to the order's laws, it is being kept secret so that no outside forces can push for any candidate in particular.
Delegates are expected to chose their new leader by January 21, when they are scheduled to be received by Pope Benedict.
The German pontiff will already know who the leader is to be because the rules laid down by the order's founder say the pope must be the first person to be informed. He will be told by telephone.
After electing their leader, Jesuit delegates will stay in Rome to discuss the direction the order should take in coming years.
As Father Kolvenbach pointed out, this is closely bound up with the identity of his successor: ''By choosing one of the thousands of Jesuits capable of being superior general, the Society is saying what it wants from the future: a prophet or a wise man, an innovator or a moderate, a contemplative soul or a man of action''.
The former leader of Jesuits in Italy, Vincenzo Liberti, said the new leader should be ''in continuity with Father Kolvenbach and oriented towards strengthening the missionary spirit of the order''.
