Cat

A cat was a type of predatory and typically small feline mammal. Cats came in a variety of colours and shapes. Cats could be found on many planets and were commonly domesticated as pets. (NA: Human Nature)

Earth cats
There were many species of cats on Earth, including 'common' house cats, tigers and sabre-toothed tigers. The cats of Earth went extinct before the year 2157. (NA: Lucifer Rising) Tigers died out before 2148. (NA: St Anthony's Fire) The Tenth Doctor once claimed the cats on Earth weren't native to the planet but were aliens. Rather than the prosaic names humans tended to give them, their actual names were much grander like "Great Hunter Grimalkin Greycoat Lionheart". Donna believed he was pulling her leg. (IDW: Autopia)

Gallifreyan cats
Cats were also found on Gallifrey. (NA: Human Nature) They had been introduced into Gallifrey's eco-system by the First Doctor and Ruath. (MA: Goth Opera) One of the Presidents of Gallifrey had a pet cat. It was eaten by a mouse that had been enlarged by the Rani in an experiment. (DW: The Mark of the Rani)

Cats and the Doctor
The Doctor's relationship with cats varied.

First Doctor

 * A cat tried to attack the Doctor and his companions when they were shrunk to a small size. (DW: Planet of Giants)

Fourth Doctor

 * The Doctor used the story of Puss in Boots to befuddle a P-series Genesis III. He claimed the computer was unable to comprehend a talking cat. (DWAN: Inter-Galactic Cat)

Fifth Doctor

 * The Doctor found a cat while searching for a Kaled mutant on Earth. (DW: Resurrection of the Daleks)
 * Erimem brought a stray cat named Antranak onto the Doctor's TARDIS. (BFA: The Eye of the Scorpion) The Doctor disdained it. (BFA: No Place Like Home, Nekromanteia)

Sixth Doctor

 * This Doctor usually wore a coat with one of several cat brooches on its left lapel, and which occasionally had a variable number of cat-shaped patches sewn into its lining. He sometimes touched his cat brooch for luck. (DW: Attack of the Cybermen) One of his cat badges seemed to change its expression to match the Doctor's mood. (WC: Real Time)
 * Whilst fighting off a transformation into an Androgum in 20th century Spain, the Doctor noticed a stray cat and was tempted to eat it. (DW: The Two Doctors)

Seventh Doctor

 * The Doctor was given a pet cat named Wolsey by Joan Redfern. (NA: Human Nature)

Eighth Doctor

 * The Doctor gave Wolsey to Bernice Summerfield. (NA: The Dying Days)
 * The Doctor enjoyed playing with and talking to Carolyn McConnell's pet cat Mina. (EDA: Vampire Science)
 * The Doctor was given a tabby cat by Donna. (EDA: Legacy of the Daleks)
 * When the Eighth Doctor was infected by Zagreus, among the avatars manifested by the Doctor's TARDIS to aid him was a Cheshire Cat straight out of Alice in Wonderland. (BFA: Zagreus)
 * The Doctor once met a cat who had been given the power of thought by the Bestiary of Legendary and Magical Creatures. (ST: Phoenix)

Ninth Doctor

 * The Doctor picked up and began talking to a cat he found in 1941 London. (DW: The Empty Child)

Tenth Doctor

 * On encountering a cat, the Doctor declared he was not a cat person after his recent experiences with the Sisters of Plenitude. Chloe Webber later drew the cat, trapping it in a picture for the Isolus to play with; along with the Doctor and all the other people and things in Chloe's drawings, it was restored by Rose Tyler. (DW: Fear Her)
 * Despite this, the Doctor seemed to have no qualms with petting the kitten offspring of Thomas and Valerie Brannigan. (DW: Gridlock)
 * The Doctor and Rose encountered a cat called Mitzi, a test subject which had been sent into Hyperspace and ended up ruling the planet Photris. After freeing the humans on Photris from Mitzi's control, the Doctor and Rose took Mitzi to the Powell Estate where she would be found and adopted by a younger Rose and Jackie Tyler. (DWS: The Cat Came Back)
 * The Hervoken used the Necris to force cats to attack the Doctor. (NSA: Forever Autumn)

Eleventh Doctor

 * The materialisation of the TARDIS in 1890 France scared away a cat which had been resting in the alley. (DW: Vincent and the Doctor)
 * The Doctor psychically conversed with a house cat to glean information about what was going on in 79B Aickman Road. (DW: The Lodger) Amy Pond stroked the same house cat. (DW: The Big Bang)
 * The Doctor mentioned "a basket of kittens" as one of the things people could think about to avoid experiencing fear which would attract the Minotaur. (DW: The God Complex)

Other cats

 * Frobisher took on a job to find a domestic cat of unknown variant. (BFA: The Maltese Penguin)
 * Jasper was a sapient cartoon cat, manifested from the mind of a young girl on the Crooked World. (EDA: The Crooked World)
 * Amy Pond's favourite childhood cat was called "Biggles." (DW: The Girl Who Waited)

Behind the scenes

 * The non-canon video game Doctor Who and the Mines of Terror portrayed the Sixth Doctor travelling with a robot cat called Splinx.
 * An alternate Ninth Doctor once met a homeless cat lady. Upon noticing one dead, he said the cat "must have used up his nine lives, rather like me". (WC: Scream of the Shalka)