How long have the Jesuits been around?

Answer 1

"catholic" actually means 'universal' - the entirety of the church. So in the correct meaning of the word, the catholic church has been around ever since the the Church was founded two millennia ago.

The Church in Rome was founded very early, already existing before being visited by any of the Apostles. As the heart of the Roman Empire, with a thriving Jewish community (that was exiled twice during the Apostolic era, before being allowed to return), travellers would have brought the Gospel from Judea very quickly - Jewish residents of Rome travelling to Jerusualem for Passover could well have been present at Pentecost.

Being under the direct eye of the emperor, the church in Rome was under constant threat of persecution - all of the early bishops were martyred - and seems to have been quite disorganised. Ignatius of Antioch sends a letter to the "largest of the churches in Rome", a line that was deliberately mistranslated to the "head of the churches in Rome" until recently.

The early church in Rome was well respected though; Bishop Clement of Rome wrote to the church in Corinthians a letter that is filled with holiness and is still studied by scholars and layman alike.

Unfortunately, a side effect of the persecution of Christians was that genuine criminals who had been branded, maimed or otherwise obviously marked as criminals, would 'convert' to Christianity in the hope that their punishments would be mistaken for proof that they had suffered for the faith. These scoundrels caused numerous problems for the church for over two hundred years, culminating in AD 366 in a violent attack when Damasus claimed the title of Bishop of Rome, driving Ursinus (a deacon in service to the previous Bishop, and the claimant elected by the common people) out of the city and killing many of his supporters. The riots and fighting resulted in the eventual decision to forbid the laity from voting; Damasus' claim to higher status than other bishops was lambasted by the scholar Jerome. However, by shrewd politicking Damasus secured the support of the emperor and by loudly opposing the major heresy of the day (Arianism) gained the grudging acceptance. Modern Roman Catholicism dates from Damasus, despite several breaks in the papal chain since.

Answer 2

Catholic Answer
St. Ignatius of Loyola founded the Jesuits in 1534, and it was approved by Pope Paul III in 1540.
from Modern Catholic Dictionary by John A. Hardon, S.J. Doubleday & Co., Inc. Garden City, NY 1980
Jesuits. The Society of Jesus, founded by St. Ignatius Loyola and approved by Pope Paul III in 1540. As conceived by the founder, it had a twofold aim: to strengthen and where necessary to restore the Catholic faith in the wake of the Protestant Reformation, and to preach the Gospel in non-Christian lands. Typical of the first purpose was the establishment of colleges throughout Europe, and the second purpose was the development of worldwide mission enterprises in Asia, Africa, and the newly developed Americas.
The Society of Jesus grew out of the Spiritual Exercises of St. Ignatius, and its structure and discipline were embodied in the detailed Constitutions, also written by the founder.