What Do Buddhists Mean by Lineage?
Lineage is a word that comes up a lot in discussions of Buddhism, but the word is not always used to mean the same thing. What are Buddhists talking about when they are talking about lineage?
There are ordination lineages and teacher/transmission lineages, which are especially important in Zen. Sometimes in Tibetan Buddhism "lineage" refers to the lineage of reborn masters. Let's take these one at a time.
Monk and Nun Ordination Lineages
Ordinations are the ceremonies confirming that a man or woman has entered the monastic orders. The many schools of Buddhism have different rules and orders of monasticism, but generally there are two levels of ordination, novice and full. Fully ordained nuns and monks also function as priests. They have full authority to give teachings and sermons and preside at ceremonies.
The procedure for ordination is recorded in the Vinaya-pitaka. The historical Buddha made the rules for ordination to maintain some kind of standard for admittance to the order, especially when he couldn't be there personally.
Among the stipulations is a requirement that a certain number of fully ordained monks be present at the ordination of monks, and a certain number of fully ordained monks and nuns be present at the ordination of nuns. Adherence to this rule has created ordination lineages, meaning an unbroken line of ordination attendees going back to the historical Buddha himself. In some schools, only ordinations within a recognized lineage are considered authentic.
Buddhists also speak of ordinations according to what version of the Vinaya they are using. Because the Vinaya originally was preserved by being memorized and chanted -- in at least two languages and probably more -- some variations occurred. The three Vinaya monks' lineage traditions recognized as unbroken are the Pali (authoritative in Theravada Buddhism), the Dharmaguptaka (used by Mahayana Buddhists in East Asia, including Taiwan, China, South Korea, and Vietnam) and the Mulasarvastivada (authoritative in Tibetan Buddhism).
Only the Dharmaguptaka Vinaya tradition has an unbroken lineage of nun's ordinations.
This has created a problem for Theravada and Tibetan Buddhism, because without an unbroken ordination lineage women cannot be fully ordained under the traditional rules.
Teaching Lineages in Zen and Esoteric Buddhism
If there's one thing held sacred in Zen Buddhism, its teaching lineages. Zen has sometimes defined itself as "face to face transmission of the dharma outside the sutras," and that's taken pretty literally. Zen tradition requires students and teachers to work together, in person, usually over a period of years, until the student's intuitive, perceptual realization of the Buddha's teaching is at least equal to the teacher's,
Traditionally, "dharma transmission" is the formal recognition by a teacher that a student has surpassed him in realization of the dharma. The faith of Zen is that the unbroken line of teachers and students goes back to the historical Buddha and the Buddhas before the historical Buddha, and in this way the living mind of Buddha is transmitted through the generations. Each Zen dharma heir comes with a lineage chart that lists his teacher, and his teacher's teacher, and his teacher's teacher's teacher, going back to the Buddha.
The lineage charts are no doubt patched together in spots, and history records that occasionally dharma heirs turned out to be bozos. Still, the lineages and the teaching process surrounding the face-to-face transmission truly are the heart of Zen. People who presume to teach without formal transmission are rarely accepted by other zennies as legitimate teachers. I believe the lineage charts are accurate going back at least a thousand years, if not longer, which is nothing to sneeze at.
When we speak of esoteric Buddhism we're usually talking about Tibetan Buddhism and the Japanese sect of Shingon. In many esoteric schools there are oral teachings that are not written down and may only be received from a teacher, and the lineages of the transmission of oral teachings are carefully maintained. There also are teacher lineage traditions that are similar to the Zen tradition.
Tibetan Tulku Lineages
Finally, in Tibetan Buddhism people sometimes speak of the succession of reborn teachers (tulkus) as lineages, although this is probably the least common usage of the word "lineage." See, for example, the "The Succession of Dalai Lamas."