Tathāgatagarbha, also known as Buddha-nature, is a concept that is found in various Buddhist traditions. It is a universal concept that is present in Mahayana Buddhism, but its interpretation and understanding may vary among different schools and lineages.

In general, Tathāgatagarbha refers to the inherent potential or essence of awakening that exists within all beings. It is often described as the "seed" or "embryo" of Buddhahood, waiting to be nurtured and realized. This concept is closely related to the idea of the Buddha's teachings being applicable to all beings, regardless of their current state or level of spiritual development.

In Mahayana Buddhism, Tathāgatagarbha is primarily associated with the teachings of the Tathāgatagarbha Sutras, which include the Mahāyāna Mahāparinirvāṇa Sūtra, the Śrīmālādevī Siṃhanāda Sūtra, and the Laṅkāvatāra Sūtra. These sutras emphasize the potential for all beings to attain enlightenment and the idea that all beings possess Buddha-nature.

However, the understanding and interpretation of Tathāgatagarbha may differ among different schools and lineages within Mahayana Buddhism. For example, in the Tiantai and Huayan schools, Tathāgatagarbha is seen as the ultimate reality or the true nature of all phenomena. In the Pure Land tradition, it is seen as the Buddha-nature within Amida Buddha, which can be accessed through devotion and faith.

In Vajrayana Buddhism, Tathāgatagarbha is often equated with the concept of "primordial purity," which refers to the inherent purity and potential for enlightenment within all beings. This view is also present in the teachings of Dzogchen and Mahamudra.

In Theravada Buddhism, the concept of Buddha-nature is not explicitly mentioned, but the idea of the potential for all beings to attain enlightenment is present in the teachings on the four stages of enlightenment and the Bodhisattva path.

In conclusion, Tathāgatagarbha or Buddha-nature is a universal concept in Buddhism, present in various traditions and schools. It emphasizes the inherent potential for enlightenment within all beings and serves as a reminder of the universal applicability of the Buddha's teachings. While its interpretation and understanding may differ, the essence of this concept remains the same - the potential for all beings to attain liberation and awakening.