The Gelug (/ɡəˈluːɡ/, also Geluk; lit. 'virtuous') is the youngest of the four major schools of Tibetan Buddhism. It was founded by Je Tsongkhapa (1357–1419), a Tibetan philosopher, tantric yogi and lama and further expanded and developed by his disciples (such as Khedrup Je, Gyaltsap Je, Dulzin Drakpa Gyaltsen, and Gendün Drubpa).

The Gelug school is alternatively known as Kadam (bKa’-gdams gsar-pa), since it sees itself as a continuation of the Kadam tradition of Atisha (c. 11th century). The school of New Kadam, or New Kadampa is an offshoot of the Gelug-tradition. Furthermore, it is also called the Ganden school, after the first monastery established by Tsongkhapa. The Ganden Tripa ("Ganden Throne Holder") is the official head of the school, though its most influential political figure is the Dalai Lama ("Ocean Teacher"). Allying themselves with the Mongol Khans, the Gelug school emerged as the dominant Buddhist school in Tibet and Mongolia since the end of the 16th century (religiously and politically). Another alternative name for this tradition is the Yellow Hat school or sect.

Doctrinally, the Gelug school promotes a unique form of prasangika Madhyamaka based on the works of Tsongkhapa. According to John Powers, Tsongkhapa's work "contains a comprehensive view of Buddhist philosophy and practice that integrates sutra and tantra, analytical reasoning, and yogic meditation."

"Ganden" is the Tibetan rendition of the Sanskrit name "Tushita", the Pure land associated with Maitreya Buddha. At first, Tsongkhapa's school was called "Ganden Choluk" meaning "the Spiritual Lineage of Ganden". By taking the first syllable of 'Ganden' and the second of 'Choluk', this was abbreviated to "Galuk" and then modified to the more easily pronounced "Gelug".

The Gelug school was also called the "New Kadam", because it saw itself a revival of the Kadam school founded by Atisha.

The Gelug school was founded by Je Tsongkhapa, an eclectic Buddhist monk and yogi who traveled Tibet studying under Kadam, Sakya, Drikung Kagyu, Jonang and Nyingma teachers. These include the Sakya scholar Rendawa (1349–1412), the Drikung Thil scholar Chenga Chokyi Gyalpo, the Kadam mystic Lama Umapa, the Jonang master Bodong Chokley Namgyal and the Dzogchen master Drupchen Lekyi Dorje.

A great admirer of the Kadam school, Tsongkhapa merged the Kadam teachings of lojong (mind training) and lamrim (stages of the path) with the Vajrayana teachings of the Sakya, Kaygu and Jonang schools. He also emphasized monasticism and a strict adherence to vinaya (monastic discipline). He combined this with extensive and unique writings on madhyamaka, Buddhist epistemology, and Buddhist practice. Tsongkhapa's numerous works on philosophy and tantric practice were widely influential and they marked a turning point in the history of Tibetan Buddhist philosophy.

Tsongkhapa and his disciples founded Ganden monastery in 1409, which was followed by the founding of Drepung (1416) and Sera (in 1419), which became the "great three" Gelug monasteries (and eventually they would become some of the largest monasteries in the world). According to Sam van Schaik these Gelug centers "came to form a triumvirate of massive Gelug monasteries that would dominate the religious and political life of Central Tibet for centuries."