Telepathy

Telepathy was the psychic ability to sense and manipulate thought, which often manifested as mind-to-mind communication. The Twelfth Doctor described the dream crabs' telepathy as "mind piracy". (TV: Last Christmas)

According to the Fourth Doctor, the presence of a time fissure could enable humans to develop telepathy and precognitive powers. (TV: Image of the Fendahl)

According to the Thirteenth Doctor, many insectoid races had telepathic abilities facilitating the hive mind. (PROSE: The Secret in Vault 13)

Telepathy could take varied forms: from an exchange of images, experiences, (TV: The Three Doctors) to associated emotions and even mind-to-mind "conversation" between two or more individuals. (AUDIO: Worldwide Web) Time Lords, for example, tended to be touch-telepaths, but weren't always limited in this fashion. (TV: The Girl in the Fireplace, The Lodger, The Eaters of Light)

TARDISes such as the Doctor's Type 40 had telepathic abilities through dedicated systems known as telepathic circuits. The circuits translated thoughts (TV: Boom Town) and any language for the passengers. (TV: The End of the World, The Fires of Pompeii) When they were trying to access an old control room, the TARDIS gave Amy and Rory the password (a sequence of words associated with feelings, sights, smells etc.) to enter it, which they then had to transmit telepathically to gain entrance. (TV: The Doctor's Wife)

Telepathy was also used by the dream crabs (which were deaf and blind) to home in on their own image in someone else's brain as long as that person was "retaining them as an active memory". (TV: Last Christmas)

A rare mineral known as matricite could block telepathy. (PROSE: Synthespians™)

On Yemaya 4 in 2257, several human colonists developed telepathic and telekinetic abilities. (PROSE: SLEEPY)

The Eleventh Doctor used a headbutt to telepathically transfer a large amount of information, including general background information, to Craig in order to explain why he was there. (TV: The Lodger)

Silurian Madame Vastra and human Jenny Flint could have telepathic conversations, such as when Jenny was struggling to hold her breath and Vastra advised Jenny to share their oxygen. (TV: Deep Breath)

According to the Doctor, back when he was young, he was able to unlock doors by using telepathy, or a variation of the ability. He used this ability again when trapped within his confession dial. (TV: Heaven Sent)

The Eleventh Doctor used telepathy to leave a psychic message in Amy's subconscious so that she would know what to do when she woke in the future after having spent 2000 years in the Pandorica. (TV: The Big Bang)

The Twelfth Doctor used the ability to tell Bill Potts to wait for him by implanting the message within her subconscious. (TV: World Enough and Time) He also attempted to wipe Bill's memories with his psychic abilities, but she discerned his intentions and begged him not to, to which he obliged. (TV: The Pilot)

Fester seemed to display a level of telepathic ability, being able to communicate with other species, most notably humans, in a non-verbal way. (PROSE: The Story of Fester Cat)

Telepathic species

 * Gallifreyans (including Time Lords) (TV: The Sensorites)
 * The Master Race (TV: The End of Time)
 * Sensorites (TV: The Sensorites)
 * Humans (TV: Planet of the Spiders, The Face of Evil, PROSE: SLEEPY)
 * Eight Legs (TV: Planet of the Spiders)
 * The Xhinn (PROSE: Amorality Tale)
 * Eternals (TV: Enlightenment)
 * Psychic worms (AUDIO: The Wormery)
 * Kitlings (TV: Survival)
 * Cheetah People (TV: Survival)
 * Boekind (TV: New Earth)
 * Weevils (TV: Combat)
 * Arcateenians (TV: Greeks Bearing Gifts, Invasion of the Bane)
 * The Ood (TV: The Impossible Planet, The Satan Pit, Planet of the Ood, The End of Time)
 * Vespiforms (TV: The Unicorn and the Wasp)
 * Silurians (TV: Doctor Who and the Silurians, Warriors of the Deep, Deep Breath)
 * Haemovores (TV: The Curse of Fenric)
 * Osirans (TV: Pyramids of Mars)
 * Nestenes (TV: Spearhead from Space, Terror of the Autons, Rose)
 * Bodach (TV: Dream-Eaters)
 * Bane (TV: Invasion of the Bane)
 * Gastropods (TV: The Twin Dilemma)
 * Jukstans (PROSE: The Vampires of Crellium)

Individuals with telepathic ability

 * Susan Foreman (TV: The Sensorites)
 * The Doctor (AUDIO: Relative Dimensions, TV: The Girl in the Fireplace, Fear Her, Planet of the Ood, The Eleventh Hour, The Lodger, The Big Bang, The Almost People, Into the Dalek)
 * Timothy Latimer (TV: Human Nature, The Family of Blood)
 * Carmen (TV: Planet of the Dead)
 * Gwyneth (TV: The Unquiet Dead)
 * Face of Boe (TV: New Earth, Gridlock)
 * Ood Elder (TV: The End of Time)
 * Mary (TV: Greeks Bearing Gifts)
 * The Attendant (PROSE: The Secret in Vault 13)
 * Fester Cat (PROSE: The Story of Fester Cat)

Devices to facilitate telepathy

 * IRIS Machine (TV: Planet of the Spiders)
 * Telepathy pendant (TV: Greeks Bearing Gifts)