Traditionally, the suttas are divided into a number of collections, of which the first is called the Digha Nikaya. Of the suttas in this collection, the second is called the Samaññaphala Sutta... which can be translated as "A Discourse on the Fruits of Contemplative Life". In it, the Buddha describes the various advanced states of consciousness that can be attained through assiduous meditation. Amongst these is Abhiñña (usually translated as "direct knowledge") -- whose Fortean relevance will become apparent when it is broken down into its constituent components:
1. Supernormal Powers
a. Having been one he becomes many; having been many he becomes one.
b. He appears; he disappears.
c. He goes unimpeded through walls as if through space.
d. He dives in and out of the earth as if it were water.
e. He walks on water as if it were dry land.
f. Sitting cross-legged he flies through the air like a bird.
g. With his hand he touches the sun and moon.
h. He exercises bodily influence throughout the created world.
2. The Divine Ear
- He hears subtle and coarse sounds, both far and near.
3. The Knowledge of Other Minds
- -He reads the thoughts of others, and knows their state of mind.
4. The Recollection of Past Lives
- He knows his past births, and the details of those births.
5. The Divine Eye
- He sees heavenly and earthly events, both far and near.
Of these, only 1g seems unfamiliar (and indeed ludicrous) to a modern reader. Apart from that, all the commonly accepted paranormal classifications are there: Bilocation (1a), Teleportation (1b), Astral Projection (1c/d), Levitation (1e/f), Telekinesis (1h), Clairaudience (2), Telepathy (3), Past-Life Recall (4) and Clairvoyance (5)!