(written from a Production point of view)
Doctor Who is a long-running British science fiction television program, and one of the few science fiction franchises that is older than Star Trek. 'Doctor Who' is about the adventures of a mysterious time traveler called the Doctor, a centuries-old Time Lord from the planet Gallifrey who travels through time and space with (usually Human) companions, fighting monsters and righting wrongs in a time machine/spaceship known as the TARDIS (Time And Relative Dimension(s) In Space), which originally could resemble anything, but after a malfunction of its chameleon circuit, currently resembles a 1963 blue London police public phone box. The Doctor's ability to regenerate his body has allowed the series to continually recast its leading actor, contributing to its longevity; as of 2012, 11 actors have played the role on television (a previously unknown additional incarnation was introduced in 2013), and two of those incarnations - the Eleventh (and current) Doctor (Matt Smith) and the Fourth Doctor (Tom Baker) - have met the crew of the Enterprise as part of the single licensed Star Trek crossover to date, "Assimilation²."
It debuted on the BBC on November 23, 1963, and ran continuously until 1989. When Star Trek was first shown on BBC TV in the summer of 1969, it took Doctor Who's Saturday evening time slot. In 1996, a TV movie was co-produced by the BBC and Universal Pictures as a pilot and aired on FOX.
The BBC brought the show back in 2005 on BBC One; the revived series was a direct continuation of the original series, as opposed to a reimagining, reboot, or a Star Trek: The Next Generation (however, it does share similarities with TNG in that the revival has featured "next-gen" equivalents of organizations such as UNIT and descendents of characters from the original series (or original-series characters, now seen later in life, such as Sarah Jane Smith and Brigadier Lethbridge-Stewart, seen in old age much as Leonard McCoy and Spock were seen in their later years in TNG). The show has spawned three spin-off series (Torchwood, The Sarah Jane Adventures and the non-BBC spinoff K-9). K-9 and Company was developed as a pilot in 1981 but not developed further.
In addition, the BBC has also licensed the production of over 150 original audio dramas based on the series since the 1990s, and hundreds of original novels and short stories have also been published since the early 1990s; Doctor Who is the only TV-based franchise to rival Star Trek in the realm of "expanded universe" releases of this nature (in one example, similar to how Star Trek fans are known as Trekkies, Doctor Who fans are known as Whovians.)
Doctor Who references in Star Trek
Star Trek has referenced Doctor Who on a few occasions. In "The Naked Now" Riker asks for a sonic driver to open a force field, a reference to the sonic screwdriver used by the Doctor to open doors and force fields throughout the Doctor Who franchise. In "The Neutral Zone", an on-screen graphic of Clare Raymond's family listed William Hartnell, Patrick Troughton, Jon Pertwee, Tom Baker, Peter Davidson, and Colin Baker among her descendants. These are the first six actors who played the role of the Doctor on Doctor Who. ("Peter Davidson" is a misspelling of the name of Fifth Doctor actor Peter Davison.) At the time that "The Neutral Zone" was filmed, Sylvester McCoy had been cast as the Seventh Doctor, but few of his stories had been aired in the United States.
This on-screen graphic has been removed from the Blu-Ray release of this story and the actors' names (as well as those of Kermit T. Frog, Miss Piggy, Lou Grant and Mary Richards) and replaced with a much more realistic family tree for Clare Reymond's descendents.
The time-travel pod encountered by the Enterprise NX-01 in "Future Tense" was influenced by the TARDIS, the time machine from Doctor Who. Specifically, the interior of the TARDIS is larger than its exterior (or "dimensionally transcendental"), and the time-travel pod was also bigger on the inside than on the outside. The exterior of the TARDIS is in the shape of a British police box and Mike Sussman, co-writer of "Future Tense", noted: "My idea of the ship morphing into a police call box was immediately nixed by the producers!" (Star Trek Monthly issue 108)
As long-running science fiction franchises, both share many similarities which are not specific references. There is a similarity between the Borg and the Cybermen, the popular race of emotionless cyborg monsters who debuted in the 1966 serial "The Tenth Planet", and likewise between the Klingons and the warrior race the Sontarans. The Heirarchy species also bear strong physical resemblance to the Sontarans. Both also feature major characters known only as "the Doctor" and a time vortex as a method of time travel. In 2013, the episode "The Name of the Doctor" introduced a previously unknown, apparently suppressed incarnation of the Doctor, much as DS9 revealed Joran Dax, a suppressed past "incarnation" of Dax in "Equilibrium".
The Star Trek novel Ishmael by Barbara Hambly contains several references to Doctor Who: the Fourth Doctor is described on page 13, Metebelis crystals (from "The Green Death", "Planet of the Spiders" amd "Hide" are mentioned on page 57, the Second Doctor is described on page 154, and Kirk recalls legends of a planet of stagnant time-travelers (meaning the Doctor's people, the Time Lords) in the Kasterborous galaxy on page 200.
The Star Trek novel How Much for Just the Planet? by John M. Ford contains a character who is not named but bears a striking similarity to Brigadier Alistair Lethbridge-Stewart in the way he talks and acts. He even orders a character called Sergeant Benson around (possibly a reference to Sergeant Benton).
The Star Trek: New Frontier novel Blind Man's Bluff features Seven of Nine introducing Soleta to The Doctor (Voyager's EMH). Soleta replies that she once met an odd man in a long brown coat and blue suit who called himself "the Doctor," a reference to the tenth incarnation of the Doctor. Later, Voyager's Doctor uses the Tenth Doctor's catchphrase "I'm sorry. I'm so sorry." In addition, a line of dialogue from later in the book includes "Don't look away... don't blink... if you blink...", which is likely a meta-reference to the Weeping Angels introduced in the Doctor Who episode "Blink".
The Star Trek: Department of Temporal Investigations novel Watching the Clock features the DTI vault that contains all the confiscated time machines. One time machine is described as a big blue box, a reference to the TARDIS.
Star Trek references in Doctor Who
The only direct reference to Star Trek in the original 1963–1989 series of Doctor Who was a panel in the TARDIS console room used in Season Fourteen that had a stained glass representation of the Star Trek insignia. However, it occasionally did feature concepts similar to those found in Star Trek. Some of these similarities were due to coincidence, or due to both series drawing from a common pool of science fiction tropes and concepts.
The 1966 serial "The Power of the Daleks" took place on a rocky planet named Vulcan. Based on the dates when this story was written and aired, this almost certainly happened by coincidence.
From the late 1960s onward, Doctor Who has featured technology similar to the transporter in the form of transmat. Transmat technology first appeared in "The Mutants" and later in several other stories. Transmat differs from transporter technology in that it does not require a trained operator to use. (The 1969 story "The Seeds of Death" had featured travel-mat technology, but this had more differences.)
The 1972 story "The Curse of Peladon" and its 1974 sequel "The Monster of Peladon", both set in the future, featured a Galactic Federation not unlike the UFP; the BBC's guide to "The Curse of Peladon" notes its roots in Star Trek, particularly "Journey to Babel". 
1996 TV movie
In the classic television series, the mechanism that enables the TARDIS to take the form of a police box is usually called a "chameleon circuit". However, in the 1996 BBC/Universal co-produced TV movie, the Doctor refers to the mechanism as a "cloaking device".
2005 – present series
There have been at least three direct references to Star Trek in the 21st century Doctor Who revival. In the episode "The Empty Child", Rose Tyler expressed dismay at the Ninth Doctor's low-tech approach to problem-solving (for example, asking questions instead of scanning for alien tech) and says, "Give me some Spock!" Later in the same episode, Rose introduced the Doctor to Captain Jack Harkness as "Mr. Spock", and Harkness briefly referred to the Doctor by this name before being corrected. In "Fear Her", the Doctor taught a child the Vulcan salute. He also named warp drive as one of the few things you need to travel the universe. In the 2011 episode "The Impossible Astronaut", when a woman from 1969 first sees one of the Silence, she asks, "Is that a mask? Is that a Star Trek thing?" and shortly afterwards, having instantly forgotten it as soon as she looked away from it, repeats, "Is that a Star Trek mask?"
There have also been several indirect references and homages. In "Flesh and Stone", the Doctor calls the 51st century starliner Byzantium a Galaxy-class ship, apparently a reference to the USS Enterprise-D from Star Trek: The Next Generation, which is a ship of that class. In the same episode, River Song states that "they'll beam me up any second." A starliner of the same class, again called a Galaxy-class ship in dialogue, appears in the 2010 Doctor Who Christmas special "A Christmas Carol". The bridge of the ship has a similar layout and design to the bridge of the USS Enterprise from the 2009 film Star Trek. The bridge scenes featured a high amount of lens flare equally reminiscent of the bridge scenes in the film. (However, it should be noted that a Draconian battle cruiser was first referred to as a Galaxy-class ship as early as the 1973 Doctor Who serial "Frontier in Space".)
In "The Lodger", the Doctor, while standing in an alien spaceship and talking to an Emergency Hologram, says, "Please state the nature of the emergency," which is almost the exact catchphrase of The Doctor, the EMH aboard USS Voyager, albeit this may have been a reference to Thunderbirds and not Star Trek, as the Doctor mentions International Rescue in the same line.
The Doctor also shares some similarities to Vulcans. The Third Doctor often performed the equivalent of a Vulcan nerve pinch and, on several occasions, both the Tenth Doctor and his rival, the Master, each performed a telepathic link very similar to a mind meld. Also, as a Time Lord, he is similar to El-Aurians such as Guinan in that he is time- and space-sensitive. Furthermore, Gallifreyan attitudes towards non-interference in primitive cultures (illustrated from "The War Games" onward) mirror not only the Prime Directive but are also criticized in similar fashion to Vulcan attitudes shown throughout Star Trek: Enterprise.
A space station of the same design as the Spacedock-type first seen in 1985's Star Trek III: The Search for Spock appeared on the front cover of the 1986 Doctor Who novel Turlough and the Earthlink Dilemma. The artwork for the novel was modified from a piece that appeared on the back cover of the LaserDisc release of Star Trek III: The Search for Spock.
"I've seen lots of Star Trek: The Next Generation, I think it's a lovely show – but there's one episode, the billing for which is so fascinating I've actively avoided ever seeing it," Davies explained. "I love the idea so much, I'd rather think about it. Forever. The episode is called 'Darmok', and the synopsis simply says that Captain Picard is trapped on a planet with an alien who can only talk in metaphors. Wow. That sounds brilliant. How does that work? What happens? How does it end? I've got no idea – not seen it! But it keeps resonating with me. I've just looked up its TX date, and it's almost 20 years old. I've been thinking about that story and its potential for almost 20 years! Would it have sustained itself for that long in my head if I'd seen it on BBC2, long ago in 1991? I think the mystery keeps the concept alive. Here I am, still wondering, right now! And I can see the idea bleeding into my own work. In 2008, I wrote a Doctor Who episode called 'Midnight'. Is it like 'Darmok'? I don't know. But stripped down to its essentials, it's a story about a hero, an alien, and words. That's practically the same billing. Maybe the two shows are profoundly different, but I know for a fact that all those years of wondering about 'Darmok' led me to that script." (SFX, issue #200, p. 140)
The Sarah Jane Adventures spin-off
Star Trek has been referenced three times in the television spin-off series The Sarah Jane Adventures. In "Warriors of Kudlak", when Luke Smith asks Clyde Langer for his cell phone while aboard a spaceship, another character scoffs that it will be useless in space unless he knows Captain Kirk's phone number. In "Mona Lisa's Revenge", Clyde describes Luke as being "all science and logic and Spocky stuff like that". And in "The Empty Planet", Rani describes her own conversation with Clyde about aliens as "talking all Star Trek".
Star Trek has been referenced twice in the television spin-off series K9. In "The Bounty Hunter", a news broadcast mentioned an experimental spacecraft called NX-2000 which underwent first flight tests. In "Jaws of Orthrus", when a CCPC pursued Darius Pike, Darius said "Resistance is Futile", the catchphrase of the Borg.
Star Trek has been referred to several times in original Doctor Who novels; for example, in the novel The Left-Handed Hummingbird by Kate Orman, the Doctor's companion Bernice Summerfield says that the first time she saw Star Trek, she thought it was a documentary. The novel The Blue Angel by Paul Magrs has an extended pastiche of Star Trek with analogues of Captain Kirk, the Enterprise and the Federation. The audio play Bang-Bang-A-Boom! by Gareth Roberts and Clayton Hickman also pastiches Star Trek, taking place on a space station called "Dark Space 8" and featuring supporting characters and events spoofing various elements of Trek-style science fiction.
In the audiobook Pest Control by Peter Anghelides, the Doctor's companion Donna Noble gives herself and the Doctor aliases based on Star Trek characters. Specifically, she dubs the Doctor "Dr. McCoy" and takes the alias of "Capt. Kirk" for herself, and is in fact referred to by that name on several occasions until the ruse is discovered.
The illustration for the Doctor Who Magazine short story, "The Useful Pile", clearly shows a Starfleet uniform from the later TNG seasons — complete with combadge — hanging in the newly-regenerated Seventh Doctor's TARDIS wardrobe.
In 2009, Russell T Davies, the writer/producer who was responsible for Doctor Who from 2005 through 2009, told an interviewer for The Times of London:
- "I would have loved to have done a Star Trek crossover. The very first year, we talked about it. Then Star Trek finally went off air. Landing the TARDIS on board the Enterprise would have been magnificent. Can you imagine what their script department would have wanted, and what I would have wanted? It would have been the biggest battle." 
Although the vast nature of both franchises, as well as the various international distribution rights owned by multiple companies, mean it is unlikely an on-screen crossover could ever be produced, IDW Publishing published a crossover comic, "Assimilation²", in 2012.
Actors who have appeared in both franchises
Numerous actors have had credited roles for episodes and/or films set in both the Star Trek and Doctor Who franchises. Prior to 2011, only a relative few actors had appeared in both. However, this number expanded considerably as a result of the Torchwood season titled Miracle Day, which was partially filmed in the US.
- ↑ The filming of "Shada" was interrupted by a strike, and the serial was never completed; however, a partial reconstruction of the story with linking narration was later released on video.
- ↑ This is a seven-part animated serial which aired in six parts in late 2009 on the BBC's Red Button service and later in one part on BBC 2.
Several other actors, including Alexander Siddig, Benedict Cumberbatch, Alec Newman, Chase Masterson, Ed Bishop, and Alice Krige as well as some of those listed above such as David Warner and Daphne Ashbrook, have also performed in audio dramas based on Doctor Who and produced by Big Finish Productions. (Warner, in fact, portrayed an out-of-continuity alternative version of the Doctor in two audio dramas, "Sympathy for the Devil" and "Masters of War"). Though licensed by the BBC, the audios exist in a gray area of canon as, unlike Paramount Pictures and Star Trek, the BBC has never decreed what constitutes canon in Doctor Who. Simon Pegg has participated in Big Finish audio dramas, and has also served as narrator for Doctor Who Confidential, a behind-the-scenes documentary series the BBC airs in conjunction with the main program. Chase Masterson has the distinction of being the first Trek actor to actually lead an expanded Doctor Who universe spin-off when she was cast as Vienna Salvatori for a Seventh Doctor Big Finish audio drama in 2012, which led to the commissioning of a spin-off audio series, Vienna, focusing on the character.
Furthermore, Patrick Stewart was reportedly one of those considered to be the Eighth Doctor - the incarnation featured in the 1996 TV movie - before the part went to Paul McGann, but Stewart has denied that he was approached in interviews. Stewart has also starred in productions of Hamlet alongside Lalla Ward (Princess Astra and Romana) and also David Tennant (the Tenth Doctor.) Ironically, Ward has talked in interviews about Stewart ribbing her for starrring in a hit sci-fi series.
Production personnel who have worked on both franchises
Ron Thornton of Foundation Imaging began his special effects career building physical models (uncredited) for Doctor Who;  he later worked as visual effects producer for several episodes of Star Trek: Voyager and Star Trek: Enterprise and the Director's Edition of Star Trek: The Motion Picture, as well as providing additional 3D matte elements for Star Trek Nemesis.
Composer John Debney, who composed music for episodes of Star Trek: The Next Generation and Star Trek: Deep Space Nine, also composed the score for the 1996 Doctor Who TV movie co-produced by Universal Pictures and the BBC for Fox Television.
Tony Dow, who directed the Star Trek: Deep Space Nine episode "Field of Fire", was the visual effects producer for the aforementioned Doctor Who TV movie. Eric Alba, who worked as a visual effects associate on Star Trek: The Next Generation, Star Trek: Deep Space Nine, and Star Trek: Voyager under the name Frederick G. Alba, was the visual effects supervisor for the Doctor Who TV movie.
David Wise, who co-wrote the Animated Series episode "How Sharper Than a Serpent's Tooth", wrote "Forever", an episode of the Big Finish Productions audio drama series Gallifrey released in March 2011.
Star Trek: Deep Space Nine writer Jane Espenson and Star Trek: Enterprise writer John Shiban have written episodes for Torchwood: Miracle Day, the fourth season of Torchwood, making them the first writers to contribute to both the Star Trek and Doctor Who franchises on television. 
Kelly A. Manners, who worked as Unit Production Manager on the Star Trek: The Next Generation episode "Coming of Age", is the Producer of Torchwood: Miracle Day. Other Star Trek production alumni who worked on Miracle Day include make-up artist Todd McIntosh and hairstylist Susan Boyd.
During a period in the 1990s when plans for a Doctor Who feature film were being bounced around by different parties, Leonard Nimoy was mooted as a potential director for such a movie.