Nachman died of tuberculosis in Uman in 1810, at thirty-eight, and his followers appointed nobody in his place. In a movement organised entirely around the living rebbe that is a structural anomaly, and outsiders named the group for it, calling them the Hasidim of the dead rebbe. Breslov Hasidim have generally accepted the description and reversed its force, holding that the rebbe remains the rebbe and that the absence of a successor is a condition of the group rather than a failure of it.
The consequence is that Breslov has no court and no dynasty, and therefore no single authority. It exists as a set of study circles, publishers, synagogues and teachers, some in strict Haredi communities and some among people with no other religious affiliation, who agree about the books and about very little else. That looseness is why it grew so widely in Israel and beyond in the later twentieth century, and why the group is difficult to count.
The one thing that gathers it is the Rosh Hashanah pilgrimage. Nachman asked his followers to be with him for the new year, and they have gone to his grave in Uman to do it, in numbers that now run to tens of thousands, through a Soviet period when reaching the town at all was dangerous and required considerable ingenuity. The Tikkun HaKlali, ten psalms he specified as a general remedy, is recited there and daily elsewhere.