Precognition

Precognition, also referred to premonition, was the ability to perceive events which had not yet happened, sometimes in prophecies.

The Time Lord Engin declared precognition "impossible" after the Fourth Doctor told him of his precognitive vision of the assassination of the Lord President. (TV: The Deadly Assassin) Long before the War in Heaven, the Great Houses used precognitive cat people as oracles. (PROSE: The Return of the King) By the Presidency of Romana III, the Celestial Intervention Agency maintained a permanent staff of precogs. (AUDIO: Intervention Earth)

As with other "paranormal" phenomena, the presence of a time fissure could enable humans to develop precognitive powers. The Fourth Doctor attributed the special powers possessed by Ma Tyler (respected by locals as a witch) to the fact that she had always lived near such a time fissure. (TV: Image of the Fendahl)

One half of the quantum transducer enabled users to see the future, the other half relaying visions of past events. (TV: Ghost Machine)

The Twelfth Doctor dismissed premonitions as the result of repeated time travel, summarising them to Clara Oswald as "remembering in the wrong order". (TV: The Girl Who Died)

Instances
Cassandra foresaw the defeat of Troy in a dream, but could not get anyone to believe her. (TV: The Myth Makers) After her resurrection in the City of the Saved, she also foresaw the Civil War and the destruction of the City, but just as before, no one believed her. (PROSE: Of the City of the Saved... Prologue)

Though the matter had some ambiguity attached to it, the Second Doctor seemed to have precognitive foreknowledge of what he would encounter in the Land of Fiction. (TV: The Mind Robber)

Leela had an apparent precognitive dream of the Fendahl. (TV: Image of the Fendahl)

Gwen Cooper had a precognitive vision of the stabbing death of her boyfriend, Rhys. She managed to prevent the event in the location in which she saw his death as having occurred, but could not prevent his death from happening elsewhere. (TV: End of Days)

The soothsayers of Pompeii, due to their minds being opened up to have psychic powers and the explosion of Vesuvius opening a rift in time for a second which echoed back through time, developed precognitive powers (TV: The Fires of Pompeii) and everything they predicted came true, including the return of Rose Tyler and the Time Beetle on Donna Noble's back. (TV: Turn Left) This power disappeared after the volcano erupted — Evelina could no longer see the future as the Pyroviles were dead and the rift had passed. (TV: The Fires of Pompeii)

The Ood, who possessed telepathic abilities, showed some precognition, with Ood Sigma predicting that the Doctor's song was ending (referring to his impending regeneration into the Eleventh Doctor) and later, when time was bleeding, they were able to predict the return of the Master, Gallifrey, the Time Lords and that Joshua Naismith was important to what was going on. (TV: Planet of the Ood, The End of Time)

Carmen, who possessed psychic abilities, predicted the Doctor's coming death telling him that "it (Gallifrey) is returning" and that "he (Wilfred Mott) will knock four times." (TV: Planet of the Dead)

The people of Eve were a time-sensitive race and were able to read timelines (i.e. seeing the past and reading the future). However, they were all but exterminated in the Last Great Time War. (TV: The Mad Woman in the Attic)

The Seeker on the planet Ribos was a clairvoyant, able to foresee her own death and the death of some people in her company. The source of her power was not clear. (TV: The Ribos Operation)

Some Basprali women possessed the gift of future insight and guided Basprali society as the Seers' Union. (PROSE: The Inquisitor's Story)

Before the Fall of Arcadia, Time Lord soldiers began complaining about premonitions regarding the Fall of Gallifrey. The Time Lord War Council dismissed these as hallucinations. (TV: The Last Day, PROSE: A Brief History of Time Lords)