The Kagyu is not one institution. It is a bundle of lineages descending from Gampopa's students, and its internal structure explains both its resilience and its most public modern crisis.
The branching
Four great subschools go back to Gampopa's immediate disciples, and eight lesser ones to the students of one of them, Phagmodrupa. Lesser here counts generations, not importance: the Drikung and the Drukpa, both in the eight, are large and independent traditions, and the Drukpa is the state school of Bhutan. Several of the twelve no longer exist as separate institutions, and the Karma Kagyu is the branch most often meant when the name is used without qualification.
The invention of the tulku
The recognition of a child as the rebirth of a dead teacher, the practice that structures Tibetan religious authority to this day, begins here. The second Karmapa, in the thirteenth century, was the first figure identified this way, and every other line of recognised incarnations follows the precedent. A system designed to keep a teaching lineage intact without hereditary succession became the mechanism by which estates and offices passed as well.
The dispute
Since 1992 two men have been recognised as the seventeenth Karmapa, by different senior figures, with the Chinese state endorsing one and a substantial part of the exile community the other. The argument turns on the authenticity of a prediction letter and on who has standing to recognise. It has been through the Indian courts, it is not resolved, and both men have continued to teach.