The actual production life of this series, from 1966 to 1973, was noted by the Star Trek Encyclopedia (4th ed., vol. 2, p. 47). This reference work said that "the series dealt with a group of secret agents who engaged in extralegal adventures on behalf of their government."
Aside from the direct reference in the Star Trek: Voyager episode, Gene Roddenberry's Star Trek and Bruce Geller's Mission: Impossible have shared many, predominently behind-the-scenes, connections throughout the years. Being simulteneously developed, and produced by the same production company, made the two series de facto franchise twin siblings. And as franchise siblings they shared a similar history of downfall and resurgence.
Like Star Trek: The Original Series, Mission: Impossible was produced at Desilu Studios, starting regular series production at the same time in March 1966, after both series were acquired by executives Oscar Katz and Herb Solow on behalf of their employer in the spring of 1964. (Inside Star Trek: The Real Story, 1997, pp. 6-7, 103) The latter show was filmed on Stages 7 and 8, while the former was filmed on Stages 9 and 10. (Encyclopedia (4th ed., vol. 2, p. 47)) Having had a first-refusal agreement with Desilu, Mission: Impossible was aired by CBS Broadcasting, which, for its own reasons, had declined to pick up Star Trek in late April 1964. That show went shortly thereafter to broadcaster NBC.
For the pilot episode of this series, Robert Justman was the associated producer and Matt Jefferies its art director. (Encyclopedia (4th ed., vol. 2, p. 47)) The pilot was produced in the spring of 1965, inbetween the first and second Star Trek pilot, both of which served in the same capacity by Jefferies and who continued to do so afterwards, whereas Justman started his long Star Trek association as such with the second. (Star Trek: The Original Series Sketchbook, p. 114) Jefferies' superior, Supervising Art Director Rolland M. Brooks, worked on both shows in the timespan 1965-1967, as did post-production staffer Craig Thompson, he until 1969. Alexander Singer became one of the series' guest directors.
Even though already picked up by their respective broadcasters, with pilot episodes produced, both series were in February 1966 under cancellation advisement by the Desilu Board of Directors, who, not entirely unjustified, feared that the small ailing production company was financially overstretching itself. Vigorously defended by Herb Solow, he managed to get studio owner Lucille Ball on his side, who overruled her board, thereby saving both series. (These Are the Voyages: TOS Season One, 1st ed, pp. 32, 94) Ball's daughter, Lucie Arnaz (wife of Star Trek V: The Final Frontier actor Laurence Luckinbill), recalled in 2006, "[At one point, her own studio chiefs said], "And the two most expensive shows are Mission: Impossible and Star Trek, [so] they have to go." She used to always listen to everything the dyed-print suits said. But she said, "No, I like 'em!" And they said, "They cost too much!" And she said, "But I like 'em!" So they left them!"
While Lucille Ball had saved the series proper, she did came close to canceling its two lead performers, Barbara Bain and Martin Landau. Ball was well known for her character trait of valuing moral propriety after her failed marriage to Desi Arnaz (which had fallen apart partly due to Arnaz' philandering), and this she explicitly expected from her staff and employees as well. When she found out that the pair were actually a married couple, she wanted to fire them on the spot as she suspected a severe case of nepotism, something she could not abide. And indeed, almost at the same time she found out about a similar case on the Star Trek lot, albeit at the opposite end of the spectrum. Ball became aware that a still married Roddenberry had an illicit affair with Majel Barrett, behavior Ball abhored, which was only aggravated from Ball's point of view by the nepotism displayed, when he surreptitiously sneaked an as a blonde disguised Barrett back into the Star Trek production (as Christine Chapel) against the express wishes of NBC. Ball wanted both of them removed at once from her studio as well. Through her personal publicist and intermediary Howard McClay, Herb Solow had a tough time convincing a headstrong Ball otherwise in both cases, as Mission: Impossible too was produced under his auspices. (Inside Star Trek: The Real Story, 1997, p. 223; These Are the Voyages: TOS Season One, 1st ed, pp. 25-27)
While both series were in production at the time, some friction evolved between the two Desilu production teams, as CBS apparently treated Mission: Impossible better than NBC did the Original Series. Edward K. Milkis, associate producer on Star Trek, recalled somewhat irked, "Another television show on the Desilu lot, Mission Impossible, was relegated a bigger budget than Star Trek, even though our show was the more difficult one to produce in our minds. The deal Desilu made with CBS was better than the deal they were able to make with NBC. NBC never understood Star Trek. They moved it around to different time slots in the three seasons it was on the air on three different nights." (Cinefantastique, Vol 27 #11, pp. 88-89) Another source of chagrin was that the original Mission: Impossible series picked up eight Emmy Awards out of twenty-three nominations, whereas Star Trek did none out of thirteen nominations.
Both series along with the police series Mannix, Solow had in the meantime also taken into production, changed ownership when Desilu was sold in July 1967 to Gulf+Western which subsequently merged the studio with the television division of its subsidiary Paramount Pictures for it to become Paramount Television. And both properties also became somewhat a bone of contention for Gulf+Western owner and CEO Charles Bluhdorn, as former Desilu, but now Paramount Financial Executive Ed Holly once recalled in a post-sale conversation he had with the former:
"Just a week or so after the merger, when Bluhdorn had started seeing the cost figures, he called me in the middle of the night. All I heard was 'What did you sell me? I'm going to the poorhouse!' I said, 'Charlie, you must be looking at Star Trek and Mission: Impossible. Those shows are costing almost to the dollar what our projections showed they would cost. You and your people made the judgment that that was all right." (Desilu: The Story of Lucille Ball and Desi Arnaz, p. 298)
Bluhdorn's trepidations were alleviated though, as he found out that NBC was only too eager to cancel Star Trek as soon as possible, something that was succesfully achieved less than two years later, during the production of its third season. Perceived at the time as the more popular – and the more easier/cheaper to produce – series though, Mission: Impossible was retained for another four seasons, and was only canceled after its seventh in 1973.
Many Star Trek alumni have made appearances on Mission: Impossible, most notably Leonard Nimoy, who moved over after Star Trek was canceled, to become a regular during the series' Season Four (1969-1970) and Season Five (1970-1971), playing "The Great Paris", a master of disguise (Nimoy's character replaced Martin Landau's Rollin Hand). After appearing in the TOS episode "A Piece of the Action", William Shatner incidentally appeared in the 1971 Mission: Impossible episode "Encore", in which he played an aged gangster who thinks he has been transferred back to the 1930s.
When the series was finally canceled in 1973, Star Trek was already making a spectacular comeback in syndication, and its very first syndicater, Kaiser Broadcasting (which operated a small chain of local television stations along the West, and East Coast) fully expected a repeat performance for Mission: Impossible as well when it as the first one acquired the broadcast rights upon its cancellation as well. However, unlike Star Trek and much to their dismay, they and other initial hopefuls found out that the series performed dismally in after-the-fact syndication, and by the mid-1970s, Mission had all but disappeared from the airwaves worldwide, having to wait for its resurgence until the mid-1990s when Tom Cruise came along. Likewise, the third series Gulf+Western had acquired, Mannix (also running for seven seasons), performed reasonably well while it was in production, but has since its cancellation been relegated to become a minor footnote in television history at best, all but forgotten save by the most fanatical of television buffs. (Inside Star Trek: The Real Story, 1997, p. 417-418)
First revival attempt
From 1988 to 1990, the series was revived under the same title and featured several characters from the original series or their children. One of those characters, Grant Collier (son of Barney Collier), was played by Phil Morris. The series was produced by Paramount Television and broadcast by network ABC.
Paramount had the year previously successfully rebooted the Star Trek television franchise with Star Trek: The Next Generation, and hoped for a repeat performance of its other property as well. However, Mission failed on that occasion to make a comeback as a franchise, running for only two seasons, due to the mishandling of the series by ABC in regard to assigned time slots, eerily reminiscent of the way NBC had treated the Star Trek: The Original Series two decades earlier. Barely remembered by even Mission fans, the second series has become the least known iteration of the franchise.
Second revival attempt
Far more successful became the relaunch as a film franchise, initiated, headed and co-produced by its principal star Tom Cruise. Mission: Impossible II, the 2000 sequel to the film that successfully relaunched the franchise in 1996, was written by longtime Star Trek writers Brannon Braga and Ronald D. Moore. The set of the same film included autoclave ovens that were reused as silver wall panels with round, light blue lights in Enterprise's sickbay in Star Trek: Enterprise. ("Broken Bow" text commentary, ENT Season 1 DVD) The 2006 follow-up, Mission: Impossible III, in which Simon Pegg made his franchise debut in the major recurring supporting role as Benji, was directed and produced by Star Trek and Star Trek Into Darkness director J.J. Abrams, under the production of his company Bad Robot Productions (having, like Abrams himself, served both franchises), whereas the fourth, 2011 film in the franchise was also produced by Abrams and Bad Robot, and on which Mitch Suskin served as the VFX supervisor.
Industrial Light & Magic, Pixomondo, Double Negative and their staffers are among the companies which provided the visual effects for one or more outings in the Mission: Impossible film franchise, having likewise done so for the Star Trek live action franchise.
Incidentally, after the the somewhat disappointing performance of Mission: Impossible III, film franchise creator, producer and super star Cruise was fired by National Amusements (holding company of Viacom/Paramount) owner Sumner Redstone in person in 2006, ostentatiously for his well publicized aberrant public behavior,  and a situation very reminiscent of Lucille Ball's intentions with the two lead performers of the original series. However, like the Bain/Landau couple had with Herb Solow, Cruise had a powerful ally in Abrams, a fast rising star and Paramount's "wonder boy" at that moment in time, who was instrumental in his reinstatement as franchise head,  leading up to the three most successful installments of the film franchise in 2011, 2015 and 2018.
Both franchises were until 2006 owned by Paramount Pictures, after which ownership shifted to CBS Corporation with Paramount, like Star Trek, retaining a license to continue producing film features.  With the exception of the first and, somewhat ironically, third ones – albeit only by the slim margin of US$10 million dollar in the first case, and the more substantial US$70 million in the second – , all other Mission: Impossible films have performed markedly better than the best box-office performing alternate reality Star Trek film, Into Darkness, which managed to gross US$467 million worldwide. 
On 13 August 2013, the official announcement was made that CBS and (new) Viacom were to be remerged into the newly named ViacomCBS Inc. entity (essentially the resurrection of "old" Viacom),   which was officially effectuated on 4 December 2019 after the obligatory approval by regulators had been secured.  In the official CBS press release both Mission: Impossible and her franchise sibling were actually as the only ones specifically mentioned by name by the statement that the remerger "reunites fan-favorite franchises such as Star Trek and Mission: Impossible",  which in itself was indicative of Mission's potential return to television no longer being considered a "Mission: Impossible", as implied by new CBS CEO Joe Ianniello to investors. 
While Mission: Impossible had originally been the lesser, underachieving one of the two twin sibling franchises, its film franchise has in the mean time outperformed that of Star Trek by far, with a seventh installment in production by 2020, slated for a 2021 release by Paramount,  its temporary production suspension due to the 2020 COVID-19 pandemic notwithstanding. This is in stark contrast to the Star Trek film franchise, where a fourteenth installment refuses to come to fruition, with Simon Pegg even going as far as casting his personal doubt on the continued existence of the alternate reality or the entire Star Trek film franchise itself for that matter, when stating, "Maybe television is a better format for Star Trek. That's where it started, you know."  Ironically, Pegg seemed to have overlooked the fact that late bloomer Mission: Impossible has started out in life on television as well.
The following is a listing of the actors who have made appearances on both Star Trek and Mission: Impossible.
- Charles Macaulay
- Blaisdel Makee
- Biff Manard
- Robert Mandan
- Monte Markham
- Steven Marlo
- Scott Marlowe
- Bruce Mars
- Don Marshall
- Arlene Martel
- Nan Martin
- Lev Mailer
- Charles Maxwell
- Ed McCready
- John McLiam
- Lee Meriwether
- Dallas Mitchell
- Lawrence Montaigne
- Ricardo Montalban
- Phil Morris (2nd series)
- Byron Morrow