Two distinct subjects share this name, and almost all confusion about it comes from treating them as one.
The first is a documented Bavarian society. Adam Weishaupt, professor of canon law at Ingolstadt, founded it on 1 May 1776. Its programme was recognisably that of the radical Enlightenment: reason against clerical authority, secular ethics, the advancement of members into positions where they could influence policy. Adolph von Knigge joined in 1780 and rebuilt its degree structure on a Masonic model, which is when it grew fastest. Bavaria's elector banned secret societies by edicts beginning in 1784, the order's papers were seized and published by the state, and it was effectively finished by about 1787. It lasted roughly a decade and its membership is usually estimated in the low thousands at its peak. Richard van Dulmen's documentary study is the standard treatment.
The second is a body of conspiracy literature that begins almost immediately after the order was dead. The Abbe Augustin Barruel's Memoirs Illustrating the History of Jacobinism, published from 1797, and John Robison's Proofs of a Conspiracy of the same year, independently advanced the claim that the French Revolution had been engineered by the order acting through Freemasonry. Both were written by men with a stake in the argument, Barruel a Jesuit exiled by the Revolution, and both had immediate political utility for those seeking to explain the upheaval as a plot rather than as a consequence of French conditions.
The claim has been continuously in print ever since, attaching itself to whatever the anxiety of the period happened to be: Jacobinism, then international finance, then Bolshevism, then a world government. Richard Hofstadter used it as his leading example of what he named the paranoid style, and the point he drew from it holds. The literature does not depend on the order and is not affected by anything discovered about it, because the order is a name it uses rather than a subject it studies.
The two facts most worth stating together are that the historical society genuinely existed and was genuinely suppressed, and that it had been dissolved for a decade before the conspiracy literature about it began. Its obscurity is what made it useful. A society that had survived would have been checkable.