Tibetan histories divide their own past into an earlier spread of the teaching and a later one, and the division still organises the schools. The earlier spread belongs to the imperial period, when Songtsen Gampo and then Trisong Detsen brought Indian teachers to Tibet, Shantarakshita and Padmasambhava among them, and the first monastery was built at Samye. It ended in the ninth century with the collapse of the empire and a long gap in the record.
The later spread begins in the late tenth century, driven by translators travelling to India and by Indian masters invited back, Atisha above all. Almost every school alive today dates its lineage from that second wave. The Sakya school formed around the Khon family and the transmission of the Hevajra material. The Kagyu lineages descend from Marpa the translator through Milarepa and Gampopa and subdivided into a large number of branches. The Gelug was founded by Tsongkhapa in the late fourteenth and early fifteenth centuries on a strict reading of monastic discipline and Madhyamaka.
The Nyingma, whose name simply means the old ones, is the school that identifies with the earlier transmission, and it defined itself as a distinct tradition largely in response to the newcomers. Two further traditions complicate any tidy count: the Jonang, suppressed in central Tibet in the seventeenth century and surviving in Amdo, and Bon, which is not a Buddhist school but shares much of the same scholastic and monastic apparatus.