The lineage runs from the Indian masters Tilopa and Naropa to Marpa, a Tibetan layman who travelled to India three times to collect teachings, and from Marpa to Milarepa, the hermit whose songs are among the most widely read Tibetan literature. Milarepa taught Gampopa, a physician turned monk, and it is Gampopa who joined the two streams the school still runs on.
The first stream is the six yogas of Naropa, a set of advanced tantric practices worked with the subtle body: inner heat, the illusory body, dream, clear light, the transference of consciousness at death, and the intermediate state. These are taught within retreat, traditionally the three year retreat that remains a formal stage in Kagyu training.
The second is mahamudra, the great seal, which is a direct approach to the nature of mind. Gampopa taught a form of it that he presented as available outside the tantric system, on the basis of a teacher pointing out the nature of awareness to a prepared student. That claim was disputed. Sakya Pandita argued in the thirteenth century that mahamudra without tantric empowerment is a Chinese doctrine of sudden awakening in Indian dress, and Kagyu writers answered him at length. The exchange is one of the substantial theological arguments between Tibetan schools, and it is still cited on both sides.