For Hindus

In Hinduism - Lord Shiva, the distroyer of ignorance and illusion, resides at Kailasa (Mt. Kailash Parvat) where he sits in a state of meditation along with his wife Parvati.

The ancient Koneswaram Temple (Dakshina Kailasam of Trincomalee, Sri Lanka) lies on exactly the same longitude as Mount Kailash.

For Jains

Mt Kailash - also known as Meru Parvat or Sumeru in Jainism. Ashtapada - the mountain next to Mount Kailash is the site, where first Jain Tirthankara, Rishabhadeva attained Nirvana/Moksa (liberation).

For Buddhists

Mount Kailash is the home of Buddha Demchok - it is believed by Tantric Buddhists.
Guru Rinpoche (Padmasambhava) - number of holy sites in the region is associated with Padmasambhava, whose tantric practices in holy sites around Tibet are credited with finally establishing Buddhism as main religion in the 7th - 8th century.
It is also said that Milarepa. Champion of Tantric Buddhism, arrived in Tibet to challenge Naro-Bon-chung, champion of the Bon religion of Tibet. The two magicians engaged in a terrifying sorcerers battle, but no one was able to gain a decisive advantage. At the end, it was agreed that whoever could reach the summit of Kailash first, would win. While Nora Bon-chung sat on a magic drum and soared up the slope, Milarepa's followers were dumbfounded to see him sitting still and meditating. And when Naro Bon-chung was close to the top, Milarepa suddenly moved into action and overtook him by riding on the rays of the sun, thus won.

For Bon

A religion, predates Buddhism in Tibet, maintain that the entire region and the nine-story Swastika Mountain are the seat of all spiritual power.