Mount Kōya is the name of eight sacred mountains in the Wakayama Prefecture circa fifty kilometres South of Osaka. The site and its surroundings were settled in 819 and are now known as the headquarters of the Shingon sect of Japanese Buddhism. Koyasan is part of a pilgrimage route through the Kii Mountains on the Kii Peninsula. The UNESCO declared the area a World Heritage Site in 2004 and says it "reflect['s] a persistent and extraordinarily well-documented tradition of sacred mountains over 1,200 years". The former monastery is now the town of Koyasan and it gives pilgrims and tourists the unique opportunity to lodge in one of the over one hundred Buddhist temples.
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To get there take the Nankai Kōya Line of the Nankai Electric Railway from Osaka. The last half hour is a spectacular scenic drive through a valley, steeply ascending and ending at the Gokurakubashi Station (極楽橋駅). From there the [Cable Line](Cable Line) ascends 300 metres to the 800 metre high Kōyasan Station (高野山駅) from where shuttle buses run into the town.
The sights can be easily discovered within one day, e.g. the Oku-no-In (奥の院) temple and the Kongōbu-ji (金剛峯寺) temple which are among the most famous and beautiful. If you fancy a hike there are lots of opportunities around. Other than that it is all about the experience of taking part in the life of the monks, e.g. the morning ceremony. The food in Buddhist temples in Koyasan is purely vegetarian as the monks are strict vegetarians.
For more information check out the very comprehensive sites of the Koyasan Tourist Association (Temple Lodging & Sightseeing).