The first (indeed one of the earliest science fiction RPGs) was "Star Trek: Adventure Gaming in the Final Frontier", published in 1978 to supplement its range of Star Trek miniatures. Written by a guy called Michael Scott, it was a bit basic. You can find a pdf version online but the site it's on currently causes my computer to scream malware warnings, so I'm not going to link to it. When Heritage Models lost their license, Scott produced and updated version called "Starfleet Voyages", in 1982. This was completely unlicensed and unauthorised, so did not see anything beyond the initial boxed release. It was up against the first big contender released in the same year, FASA's "Star Trek, The Role Playing Game".
I have this, picked it up earlier this year, but I've known of it for ages. This was the first big Star Trek RPG out there and it has it's fans and adherents even now. Taking a tip from Traveller the character creation procedure makes you generate a whole backstory as they move through cadethood at Starfleet Academy, have their first posting and eventually their first ship assignment. If you do really badly you end up in the Merchant Marine, which I feel is base slur on that noble institution. The first releases were set firmly in the Original and Animated series of Trek, but rapidly adopted the wider universe and styles established in the films, especially "Wrath of Khan". For its time it was a good game, but seen now as a little clunky. Personally I don't like the percentile skill system, but the game designers nevertheless gave a coherent and well thought out universe for players to explore. Of particular note was the treatment of the Klingons, drawing on the ideas of writer John M. Ford, which influenced canonical Klingons in the moves and still do to this day, despite most of the background developed for them ultimately being rejected and retconned.
Alas the advent of Star Trek, The Next Generation soured the deal, the two sourcebooks released by FASA for this being not matching their own view of how the franchise would go and in 1989 they withdrew the license. In the meantime a wholly different Star Trek universe was being developed by Task Force Games (later the Amarillo Design Bureau Inc.) who had long held permission to develop their own Star Trek derived universe for their "Star Fleet Battle"s wargame. Ultimately, in 1993, they launched their own RPG, "Prime Directive", first with their own system and then with versions tailored for Steve Jackson's GURPS RPG, both (3rd and 4th editions) and D20 Modern. This still a current system. I did have Star Fleet Battles when it first came out, then much later was enticed back to it with its rules-light cousin "Federation Commander". I also have Prime Directive D20 and it's a good product. However this is a Federation at war, at odds with the original vision of Star Trek and this was always something that sat a little uneasily with me. These is apparently a version that employs the "Traveller" rule system coming out soon, something I will be very interested in seeing, but it's been coming out "soon" for some time now so I'm not holding my breath.
New canonical Trek roleplaying came back with Last Unicorn Games, who gained a license to produce a Next Gen game in 1998, and who produced a separate Deep Space Nine game and an Original Series game in 1999, using their ICON system. I did have the first two and now have the latter, despite an Amazon seller initially shipping me a highly battered core rulebook with a separating spine, scuffed and dented cover, pages that were ripped and stuck together and, the icing on the cake, a pencilled price in the front of the book which was a sixth of what I paid for it, despite being described as "very good condition". I was not happy. A few bits of photographic proof and a rather strongly worded complaint later I had my money back and profuse apologies. I got a much more acceptable copy from somewhere else.
This system also has its fans despite a few mechanics that are particularly loathed, the so-called "drama dice" in particular, but the general consensus is that if you ignore that it's a good solid system, and it is nice to have a book that's styled to the series it deals with, so I'm quite fond of it. Alas LUG lost the license to Decipher Games in 2002, who then produced their own game, with their own "CODA" system and rulebooks that allowed you to tailor the game to suit the era you wanted. Now I don't have this and while I have seen comments from some people saying it's a superior system to Last Unicorn's, quite the majority of reviews are been rather scathing, with some going so far as to label the whole game as an unworkable, incoherent mess, mainly due to some very bad editing and rulebook layout rather than any serious complaint with the game mechanics itself it seems. Unfortunately troubles in the company forced Decipher to shut down their RPG branch in 2006 with some key source books never having been released.
Now we have a new contender with "Star Trek Adventures" from Modiphius Entertainment, one of the new and upcoming RPG companies, responsible for the most excellent "Tales From the Loop" and other. From what I've seen the game seems to have been well received, with an easily comprehended system, excellent art, high quality production and the ability to play in any Trek era (although the style of the books is firmly Next Gen in look). We will just have to see how this goes and hope it doesn't suffer the same fate as its predecessors. I haven't got this and given the rather hefty price attached I am unlikely to pick it up anytime soon.
However, this is not the whole story, and I'm not talking about the "Enterprise" RPG released only in Japan in 1983 either. How about "Starships and Spacemen" first published by Fantasy Games Unlimited in 1978? This is basically a Star Trek game with the names changed to protect the guilty, "Taurans" for Vulcans, "Zangids" for Klingons, etc., and is quite frankly not fooling anyone save apparently Paramount's lawyers. The game is now owned by the company Goblinoid Games, who have released a 2nd edition. It is a very basic game, there are only four skills, Combat, Science, Technical and Contact which you use to cover pretty much any task you might undertake, but it does capture the "feel" of Star Trek better than some of the later games (I particularly like the alien forehead shape tables). It's helped in this by the "Lucanii Drift" setting book by Paul Kidd which introduces a range of background skills for characters, adds a few of the Trek races which weren't in the rulebook and provides a sandbox setting for the characters to go boldly in.
One thing I found recently while wandering around the net was a fan made game called "Star Trek- Alpha Quadrant" that uses the Traveller/Cepheus Engine system (beating Prime Directive to the punch). It's not bad, nicely adapting the system to generate a Starfleet Career path much like FASA had and providing skills and professions that mirror the roles of the various crew members in the various eras. The author, Stephen J. Ege has clearly put a lot of work into this and it shows.
You can find it here- https://rpg.rem.uz/Traveller/00%20-%20Other%20Materials/Traveller%20Related%20Games/Cepheus%20Engine/TCE%20-%20Star%20Trek%20Alpha%20Quadrant.pdf
(This is in a HUGE online repository of RPG pdfs, the "Remuz RPG Archive", probably the largest online collection I've ever seen and which is probably in breach of any number of copyright laws.)
Finally, and saving the best for last, I spent a little while searching various RPG sites for what other people thought was the best game for Star Trek. Aside from those who recommended taking the FATE and Savage Worlds systems and adapting it to the setting, many recommended the FASA game as still one of the best, several waved the flag for Last Unicorn and fewer for Decipher, but one name kept on cropping up "Far Trek". This started life as a game using the "Microlight20" system by Mike Berkey in 2009 called "Where No Man Has Gone Before", but was subsequently taken by one C. R. Brandon and adapted to his own "Three D" system as "Far Trek". One reviewer described this as a love letter to the Original Series and he's not wrong, the whole game is geared to produce exactly the same feel as the classic and animated Star Trek gave you when you were watching it. He started work in 2011 and finally decided he'd done as much as he could do in 2016. There was a brief window where you could get a completely non-profit print version via Lulu which I am unfortunately two years late in discovering, but the material is still freely available here- http://fartrekrpg.blogspot.com/
This has the main rulebook, character sheets, some stand-up card figures to print out, an couple of adventures and conversion rules to adapt anything from the FASA Trek game. There's also a Tech Manual for the Saladin-class Destroyer recommended to put your crew on (albeit a somewhat runty shortened hull, end of production line model). Now I like this, I like it a lot, it's exactly the RPG I've been looking for to play original Trek with. In fact I liked it so much I printed the whole thing out and had it bound so I could get my own physical copy. The premise of using a Saladin-class as the player's ship was one I'd already been considering although the rules do provide the full "Franz Joseph" range of Heavy Cruiser, Destroyer, Scout, Tug and Dreadnought, with a Surya-class frigate thrown in to act as a TOS era Miranda-class Light Cruiser. The rules are the V2.1 edition and Brandon has clearly been listening to feedback, adding a number of races from the animated series like the Catian and Edoan in the appendixes in the back, together with rules for Merchants and Traders and playing Klingon and Romulan characters. Yes there are a few typos here and there and some layout errors, and the art is sadly sparse, but I'm willing to overlook these and accept it for what it is, an honest and damned successful attempt to produce the best Star Trek RPG the author could give you, a game that goes a very long way to capturing the true essence of the show.
A lot of people, both gamers and Trek fans seem to think that this is probably the best Star Trek RPG out there for playing in the original series era. I think I'm going to have to agree with them.