I'm going to answer this in two parts: questions of the existence of so-called Cathars, and then the survival of Catharism.
The Existence of Cathars
For those who do not know much about the subject, the Cathars in traditional history writing were a group of 'heretics' who it is claimed were Manichaeans, or dualist Christians. Put simply, this entails a belief in two gods: one of good and the spiritual plane, and one of evil or the material plane. Manichaeans are ascribed certain habits like disavowal of sex and procreation, not eating meat, because, apparently, this persists the 'evil' of material reality when we should be delivering ourselves to the spiritual plane. In some claims of Christian Manichaeaism thenew testament is associated with a good God, and the old testament with an evil God.
The Cathars are said to have existed in parts of eastern and western medieval Europe, but most particularly in the south of France ('Occitania') between about 1150 and 1350. It is also claimed the the Cathars were organized into formal churches which mirrored the structure of Catholic organization: preachers and bishops, even with a potential pope-like figure from somewhere in the Balkans.
These Cathars were the subject of persecution, including the Albigensian Crusades and the first papal inquisitions.
But the Cathars didn't exist.
There is no doubt that there were Christians in Occitania, and across Europe, who emphasized the new testament over the old, and who conducted preaching and some small (personal) ceremonies outside of the Catholic Church, and there were a range of disputes with certain Catholic theology and sacraments. It's not for nothing that heretics were at times also accused of Arianism and Donatism.
The foundations for claims of Cathar Manichaeaism, and the theology that goes along with it, is virtually non-existent. A few weeks ago I wrote about this specifically here and here. And without the centrality of Manichaeaism, 'Catharism' as an organized belief system, a 'religion', at least in Occitania, disappears like mist in sunlight.
The claims of a developed counter-Church prior to mid 13th c are based on a couple of deeply disputed documents which you can read me discussing in the links above. There is no evidence from inquisition records of a church. We do have some indications of some form of organization materializing after the crusades, and it seems likely a reaction to the pressures and persecutions of crusade and inquisition created a dynamic of a formal association. So, we have some contemporary mentions of deacons and priests and the like (much like the early Christian church). But again, no evidence of a separate 'Cathar' religion or belief system outside apostolic Christianity.
The name 'Cathar' itself is a term from Antiquity (catharii) that was specifically resurrected at the beginning of the 12th century by German bishop in the Rhine valley to describe a group of local heretics. With the growth of 12th c heretical expressions in communities in Germany, and more particularly France, we see Catholic writers, mostly Cistercian abbots, monks and bishops, writing fairly immense tracts against heresy and we see them visiting Occitania on counter-preaching expeditions.
Although the term was picked up, really transmitted from the Rhine Bishop into the Cistercian order, we rarely see the name Cathar used even by those same Cistercians who debated the heretics face-to-face in front of crowds at Albi and other towns. We don't see the name in more than a handful of instances in one full century that marks the high tide of high middle ages heresy and those instances are usually among collections of names. The papal bull which called the crusades against the Albigeois does not refer to Cathars. The 3 chronicles of the Albigensian Crusades, from three separate Catholic orthodox writers, do not mention Cathars. There are no uses of the term Cathar in any of the thousands of inquisition records we have from Languedoc to the Toulousaine to Quercy to the Agenais. Not even the last large inquisition into heresy in Occitania, that of Bishop Fournier of the citizens of Montaillou in the 14th century, mentions Cathars.
In fact, there isn't much to be seen of heretical theology at all among the actual experiences of the population. What we do see are expressions of apostolic Christianity, and often those strains came into conflict with Catholic orthodoxy, usually around questions of license to preach. These are the so-called 'wandering preachers'. Evidence of this is clear in the case of the contemporary Peter Waldo and the Waldensians. Catholic orthodoxy, the Popes of 13th c, attempted to embrace apostolic Christianity with licensing their own wandering preaching order, the Dominicans and the Franciscans who were to follow Francis of Assissi's model.
In fact, the term Cathar really only comes into writings after the seeming disappearance of those who were supposedly Cathars. From a historiographic point of view it looks like a retroactive application of a label to give coherence to a group of diverse beliefs and peoples.
Persistence of Catharism after crusade and inquisition
So we've now deeply problematized if not entirely dismissed the very existence of western European Cathars in (i) name, (ii) as an organized group and (iii) as a theology.
What did persists after the mid 1350's, if anything? Well, it appears that Franciscanism and various inventions of the Catholic Church around apostolic Christianity may have addressed the underlying appeals of heresy in southern France and elsewhere. Some strains of pop history like to tie reformation and Protestantism to the lineage of Catharism. Any ties would be a fiction close to a conspiracy theory. There might be some overlap in theological approaches between various Christian movements, but linking them together in a long chain is lazy history.
There are some claims to Catharism in northern Italy during the same period, but the evidence of those dies out by mid 14th century as well. Certainly Catharism or whatever it was called in Italy was not nearly as pervasive as it was in southern France.
Note: none of the foregoing reflects the history of apparent Manichaeaism and possible Catharism in parts of the Balkans, which is another story unrelated to Catholicism and the Papacy in the west.
Please let me know of any questions or thoughts.
Sources and further reading:
Moore, R.I. "When Did the Good Men of the Languedoc Become Heretics?" from the proceedings of Examining the Heretical Thought. Conference at Berkeley, California, 2006.
Moore, R.I. The War On Heresy: Faith and Power in Medieval Europe, London: Profile Books Ltd, 2012
Pegg, Mark Gregory Most Holy War: The Albigensian Crusade and the Battle for Christendom, Oxford: Oxford, 2008
Pegg, Mark Gregory The Corruption of Angels: The Great Inquisition of 1245-1246, New Jersey: Princeton, 2001