A cravat was a form of clothing worn around the neck.
Worn by the Doctor
The First Doctor wore a coal black cravat when escaping the Morok Empire's Space Museum (TV: The Space Museum), during his escape from Pursuer-Daleks as well as his first encounters with Mechonoids and Steven Taylor during the Mechonoid Incident (TV: The Chase), during his first encounter with the Monk (TV: The Time Meddler), during his first encounter with Drahvins and Rills (TV: Galaxy 4), during his first encounter with Katarina and the Sacking of Troy (TV: The Myth Makers), when foiling the Dalek Empire's master plan with the Time Destructor (TV: The Dalek's Master Plan), during his first encounter with Sontarans (AUDIO: The Sontarans), when braving the Game of Rassilon set up by President Borusa in his quest for immortality (TV: The Five Doctors), when helping his future incarnations foil the Tremas Master and witnessing Adam Mitchell's fall (COMIC: Endgame), during a trip to Paris during 1572 (TV: The Massacre), during his first adventure with Dodo Chaplet on the Ark as well as his first encounter with Monoids (TV: The Ark).
He also wore one during his first game with the Toymaker (TV: The Celestial Toymaker), when facing Pa Clanton and Johnny Ringo on 1881 (TV: The Gunfighters), during his first visit to the Der-Sen Monastery and his first encounter with Yeti (AUDIO: The Secrets of Det-Sen), when meeting Captain Edal and saying goodbye to Steven (TV: The Savages), during his confrontation with WOTAN and its War Machines as well as his farewell to Dodo (TV: The War Machines), during his first adventure with Polly and Ben in the 17th century (TV: The Smugglers), when helping his future incarnations to defeat a fleet of Mark VIII Daleks and save Gallifrey during the end of the Last Great Time War (TV: The Day of the Doctor), and right up to his fall after the Mondasian Cyberman 1986 invasion of Earth (TV: The Tenth Planet, Twice Upon A Time).
While in Paris on a mission for the Celestial Intervention Agency, the Second Doctor wore a black cravat. (PROSE: World Game)
The Third Doctor wore a number of cravats throughout his life. He wore one in ebony black, (TV: Doctor Who and the Silurians) one in midnight blue, (TV: The Ambassadors of Death) and one in vermillion. (TV: The Mutants)
The Fourth Doctor wore cravats for a period, initially adopting ones green, (TV: Terror of the Zygons) blue (TV: Planet of Evil) and salmon in colour, (TV: Pyramids of Mars) before settling on a maroon one. (TV: The Android Invasion - The Robots of Death) He also wore a ruby cravat, (TV: The Talons of Weng-Chiang) and a cravat of ivory dupioni. (TV: The Invasion of Time - The Pirate Planet)
The Sixth Doctor wore a wide variety of cravats, them being a mainstay of his outfit. His first cravat was turquoise with a polka dot pattern, (TV: The Twin Dilemma) which he eventually replaced with a similar cravat in crimson. (TV: The Mysterious Planet) During his travels with Evelyn Smythe, he briefly returned to wearing a turquoise cravat, this time without polka dots, (WC: Real Time) but soon returned to wearing cravats with such a pattern including one in periwinkle purple, (PROSE: Killing Ground) eggplant purple, (AUDIO: The Condemned) Prussian blue, (AUDIO: Project: Lazarus) and one in midnight blue. (AUDIO: The Wrong Doctor) After briefly wearing yet another polka dot cravat, this time in purple, (PROSE: The Quantum Archangel) he adopted a mustard yellow cravat with a starfield design (TV: Terror of the Vervoids) and later wore an electric blue one with pink stars. (AUDIO: Broadway Belongs to Me!) By the time of his regeneration, he had returned to wearing his crimson cravat. (TV: Time and the Rani)
The Seventh Doctor wore the Sixth Doctor's crimson cravat in his first hours before switching for a scarlet paisley tie. (TV: Time and the Rani) He also wore a black cravat with his dark jacket. (PROSE: St Anthony's Fire) During his escapades as Time's Champion, he wore a green silk cravat. (PROSE: White Darkness)
The Eighth Doctor often wore a dupioni cravat, first adopting a battleship grey cravat shortly after his regeneration. (TV: Doctor Who) He also wore some in indigo, (COMIC: Dreadnought) plum purple, (COMIC: Descendance) amber, (COMIC: Ophidius) gold, (AUDIO: The Stones of Venice) and emerald green. (AUDIO: Caerdroia). The battleship grey cravat was enshrined as a primary possession of this incarnation in a museum created within the TARDIS Matrix. {COMIC: The Forgotten)
The Ninth Doctor wore a burgundy brocade cravat in 1912 Southampton, (TV: Rose)
The Eleventh Doctor wore a black cravat on Victorian London. (TV: The Snowmen)
Sometimes, the Twelfth Doctor sometimes wore a black cravat. (TV: Mummy on the Orient Express, Thin Ice)
In his Sixth Doctor form, the Curator wore a cyan cravat with white polka dots in basic bow form. (AUDIO: The Artist at the End of Time)
Worn by Companions
Leela wore a scarlet cravat with her walking suit when she and the Fourth Doctor went to London during 1892. (TV: The Talons of Weng-Chiang)
Jenny Flint wore cravats with her waistcoats and her shirts. One was steel blue, (TV: A Good Man Goes to War) and many in ebony black. (TV: Deep Breath; PROSE: The Dangerous Dilemma of the Dream Doorway)
Bill Potts wore an ivory cravat during her trip to the 1814 frost fair. (TV: Thin Ice)
Yasmin Khan wore a cravat of bronze brown tartan during the year 1904, the 2021 Sontaran Invasion of Earth, and the Flux. (TV: Survivors of the Flux, The Vanquishers)
Josie Day wore cravats done on jade (COMIC: A Matter of Life and Death) and indigo (COMIC: The Lost Dimension)
Amy Pond wore a black and white cheque cravat with her policewoman outfit. (TV: The Eleventh Hour)
Jamie McCrimmon wore a silver cravat when he reunited with Zoe Heriot in the Memory TARDIS. (TV: The Mind Robber)
Graham O'Brien and Ryan Sinclair wore ebony black cravats when they met Mary Shelley, Lord George Gordon Byron, John Polidori, Claire Clairmont, and Ashad on 1816. (TV: The Haunting of Villa Diodati)
Jackson Lake wore a gold cravat when he and Rosita Farisi helped the Tenth Doctor to defeat Cybermen from Pete's World during the 1851 Incident. (TV: The Next Doctor)
Worn by the Master
Many incarnations of the Master wore cravats.
John Smith, a human persona of the Decayed Master, wore a black cravat. (AUDIO: Master)
When he was posing as the 'Other Doctor', the Bald Master wore cravats done in emerald green or gold. (AUDIO: Dominion)
During the Last Great Time War, the War Master wore black cravats; (AUDIO: Only the Good, The Master of Callous, Rage of the Time Lords, Day of the Master, etc.) including one in basic form when he was Professor Yana. (TV: Utopia)
Missy, a feminine version of the Master, wore a fern brocade cravat; (TV: Extremis) right up to her fall after being shot by her predecessor with a laser screwdriver to stop her from helping the Twelfth Doctor to save Mondasians from Cybermen. (TV: The Doctor Falls)
The Spy Master wore a black cravat when he met the Thirteenth Doctor on London during 1834. (TV: Spyfall)
Won by Others
Emily Morris, the 19th century girl who helped Sarah Jane, wore an ivory cravat with black spots. (TV: Lost in Time)
Professor Jericho wore cravats during the early 1900s; one in mauve and one in rust orange. (TV: Survivors of the Flux|The Vanquishers)
George Litefoot wore black cravats. (TV: The Talons of Weng-Chiang; AUDIO: Voyage to Venus)
Josiah wore a silver cravat with his Victorian outfit. (TV: Ghost Light)
Theodore Maxtible wore a bronze cravat. Edward Waterfield also wore a scarlet cravat in basic bow form. Both did this during the Dalek Empire's Operation Human Factor. (TV: The Evil of the Daleks)
John Lumic wore an onyx black cravat up to when he was converted into a Controller of the Cybus Cybermen. (TV: Rise of the Cybermen|The Age of Steel)
Webley wore a burgundy cravat on basic bow form. (TV: Nightmare in Silver)
Robert Muir wore a beige brocade cravat with a royal tan suit. He and Charles Cranleigh also wore ivory cravats Robert Muir wore an ivory cravat as part of his Georgian costume in a party round Cranleigh Hall on 1925. (TV: Black Orchid)
Rhys Williams wore an amber cravat when he married Gwen Cooper. (TV: Something Borrowed)
Sergeant Calder and his clone both wore a cravat that matched the colour of their uniforms. (TV: Resurrection of the Daleks)
Walter Simeon wore tartan cravats, (TV: The Snowmen) as did the Great Intelligence when taking his form. (TV: The Name of the Doctor)
Mr Scoones, Mr Cole, Mr Fetch and Mr Milligan wore ebony cravats up to the 1851 Incident. (TV: The Next Doctor)