Magic

Magic formed part of the fundamental "operating system" of both the universe which existed before the Doctor's own and the universe which would follow it. (PROSE: Millennial Rites) Some universes which existed "sideways in time" were also magic-based. (TV: Battlefield)

Magic was initially a strong force in the Doctor's universe during the Dark Times, with Gallifrey being ruled not by technocratic politicians but by the Pythia and her priestesses. However, as the Time Lords rose, they shaped the universe to become a clockwork set of rules which they could predict and control, with any irrational or supernatural principles replaced by those of science to underpin reality. (PROSE: Cat's Cradle: Time's Crucible, Christmas on a Rational Planet)

Had the Ferutu, rather than the Gallifreyans, become the Lords of Time, the universe would have continued along magical principles. (PROSE: Cold Fusion)

Within a rational universe
The Doctor claimed not to believe in magic (TV: The Dæmons, PROSE: The Sorcerer's Apprentice, TV: The Robots of Death) and endorsed Clarke's Law. (TV: Battlefield) Indeed, most instances of the supernatural encountered by the Doctor's first nine incarnations eventually stood revealed as the effects of alien science, be they the psychic powers and technology used by the Dæmon Azal (TV: The Dæmons), or the nanotechnology of Elbyon. (PROSE: The Sorcerer's Apprentice)

Yet the Second Doctor once stated that there was room for magic in an infinite universe. (PROSE: The Celestial Toyshop) The Seventh Doctor was not entirely inflexible on the matter, as evidenced by the corollary he added to Clarke's Law stating that any sufficiently advanced form of magic was indistinguishable from technology. (TV: Battlefield) The Eighth Doctor asserted this corollary as well, (PROSE: The People's Temple) and notably embraced the concept of magic. (PROSE: The Scarlet Empress)

The Twelfth Doctor noted "real" wrestling used, among other things, magic spells. (TV: Thin Ice)

Leela told Adelaide Lessage, "I too used to believe in magic, but the Doctor has taught me about science. It is better to believe in science." (TV: Horror of Fang Rock)

Psionics was originally a form of magic, but survived the rationalisation of the universe through being sufficiently grounded in science. Monitoring the network of connections that psionic abilities established across time allowed the Time Lords to possess an "early warning system" against outbursts of wilder magics breaking through into their ordered cosmos. (PROSE: So Vile a Sin)

When the ancient entity known as the Carnival Queen was accidentially released back into the Doctor's universe, she caused irrationality and magic to be reintroduced throughout space-time while destroying the laws of physics and nature in favor of the supernatural. The Seventh Doctor later helped repair the damage she caused, only the Carnival Queen continued to influence reality from her prison outside the universe. (PROSE: Christmas on a Rational Planet)

Hyspero was one of the few places in N-Space where magic was commonplace. (PROSE: The Scarlet Empress, The Blue Angel)

When facing off against the Kotturuh, the Tenth Doctor was vocally against labelling them as supernatural beings, noting that all they did was deploy a powerful retro-virus, even as Estinee viewed the Kotturuh as using magic spells. (PROSE: The Knight, The Fool and The Dead)

After Gallifrey
Following the first destruction of Gallifrey, (PROSE: The Ancestor Cell) the Time Lords' exorcism of the irrational from the universe was revoked and magic regained its potency. (PROSE: The City of the Dead, The Adventuress of Henrietta Street)

Following the second destruction of Gallifrey in the Last Great Time War, the Tenth Doctor was initially prepared to consider the linguistic-science of the Carrionites to be witchcraft. (TV: The Shakespeare Code)

Elsewhere, Clarke's Law continued to be the order of the day. The Sycorax' form of blood control was understood by both Daniel Llewellyn and by the Sycorax themselves to be supernatural. The Tenth Doctor's regenerative abilities were also taken for witchcraft. (TV: The Christmas Invasion)

On a number of occasions the Doctor appeared to have suggested that a magical and a scientific understanding of the same phenomenon were now interchangeable. (TV: The Girl in the Fireplace, Tooth and Claw) Others made the connection too, Elton Pope noting that the sonic screwdriver was essentially a "magic wand". (TV: Love & Monsters)

The Tenth Doctor listed spells, rituals and incantations among those things in which he personally did not believe, but remained prepared to consider that they might nevertheless be real. (TV: The Satan Pit) Jack Harkness considered some phenomena, such as fairies, to exist outside the bounds of scientific inquiry. (TV: Small Worlds)

In dreams
In a shared dream, Santa Claus claimed to Shona McCullough that his flying reindeer were a "scientific impossibility", and as a result, he fed them "magic carrots" so they could fly. (TV: Last Christmas)