Prydonian Academy

The Prydonian Academy was an educational institution on Gallifrey. Its pupils wore scarlet robes, and were Time Lords by virtue of biology, not education. Amongst other things, the Academy taught them how to instantly calculate "complex temporal induction problems". It had a fortress-like appearance and a population segmented by ability; only a select few were brought to its highest towers to be taught "darker things". (PROSE: Timewyrm: Revelation)

Physically, it was situated at the base Mount Cadon, Gallifrey's highest peak. An imposing structure, it was a dark, ebony building (PROSE: Revelation, Lucifer Rising) at the northern border of "southern Gallifrey". (Lucifer Rising)

As Lord President, the Doctor was able to access its so-called "black collection" and steal a portable temporal link. (Revelation)

People associated with the Academy
On one occasion, the Doctor claimed to have attended the Prydonian Academy with the Master, though the Doctor equivocated on the point. (PROSE: First Frontier) The Fourth Doctor, however, chose to wear Prydonian robes — after stealing them — when he visited Gallifrey on a formal occasion. (TV: The Deadly Assassin) Later, the Eighth Doctor clearly remembered having been a Prydonian on Gallifrey in his youth. (PROSE: The Face-Eater, The Eight Doctors)

According to the Eighth Doctor, Romana was a Prydonian graduate. (PROSE: The Ancestor Cell) In her third body she had a tiny Prydonian seal on her left ankle. (PROSE: The Shadows of Avalon)

Graffito was a Time Lord known to have some sort of connection to the Academy, and was famous for his or her jokes. (PROSE: Set Piece)

Ragnar was a member of the High Council sub-committee that had oversight over the Temporal Scanning Service who also was a Prydonian. (PROSE: World Game)

Opinions about the Academy
Chancellor Djarshar had great disdain for the college, saying that the institution was comprised of "renegades, fugitives, lunatics and ingrates". He went on to suggest that Prydonians "[thought] Gallifrey owed [them] everything." (PROSE: The Ancestor Cell)

Behind the scenes
Since "the Academy" typically means the "Time Lord Academy" in works like Interference - Book One and the Gallifrey audio series, and since Paul Cornell barely mentioned the place in Revelation, it's unclear whether the Prydonian Academy is a formal name for the Time Lord Academy, a part of a larger whole, or, indeed, what exactly it is. It was also unclear whether the Prydonian Academy was uniquely attached to the Prydonian Chapter in some way, because Cornell never specified any such connection. However, Russell T Davies seemed to suggest that it was merely a part of the larger educational system by creating the Academy of the Patrexes in PROSE: Damaged Goods.