The Tibet “Autonomous Region” of China borders, clockwise, Nepal, Uttarakhand state, Himachal Pradesh state, Indian Jammu and Kashmir, Aksai Chin (an almost uninhabited area in northern Kashmir claimed by both India and China: see this post), the Xinjiang Uyghur Autonomous Region of China, Qinghai province, Sichuan province, Yunnan province, Burma, Arunachal Pradesh state, Bhutan, and Sikkim state.
Aksai Chin is shown in pale green here, but so is the other area China disputes with India, northern Arunachal Pradesh, which China calls South Tibet.
The Brahmaputra, Ganges, Indus, Yellow River, Yangtze, Mekong and Salween have their sources on the Tibetan plateau.
The Brahmaputra, also called Tsangpo-Brahmaputra, begins its journey in southwestern Tibet as the Yarlung Zangbo, and flows across southern Tibet, breaking through the Himalayas in great gorges and into Arunachal Pradesh, where it is called Dihang. It flows southwest through the Assam Valley as the Brahmaputra and south through Bangladesh as the Jamuna. There it merges with the Ganges at the head of the world’s largest delta, the Sunderbans. Calcutta is on a tributary of this system, the Hooghly, and part of the Sundarbans is in India.
The Ganges rises in Uttarakhand, not technically in Tibet, flows southeast through India, and joins the Brahmaputra in Bangladesh. Its major tributary, the Jumna or Yamuna (not to be confused with Jamuna), also rises in Uttarakhand, west of the Ganges, and joins the Ganges at Allahabad.
The Indus rises in Tibet, south of Aksai Chin, and flows into Indian Kashmir, and thence into Pakistan.
The Yellow River (Huang He) rises in Qinghai, not technically in Tibet, and flows east through China, emptying into the Bohai Sea.
The Yangtze rises in Qinghai, enters China in Sichuan, flows into Yunnan, and continues east until it reaches the East China Sea at Shanghai.
The Mekong rises in Tibet and flows through Yunnan, Burma, Thailand, Laos, Cambodia and Vietnam.
The Salween rises in Tibet, and flows south into Yunnan and then into Burma (with a short passage in Thailand), where it enters the Andaman Sea.
The Ayeyarwady or Irrawaddy, not shown on the map below, flows entirely within Burma, and west of the Salween. Its source, in Kachin state, is close to Tibet.
Map above Wikimedia Commons; below from www.everythingselectric.com