I am struck by just how well-written so many of the episodes from the first two seasons of STAR TREK: THE ORIGINAL SERIES are, despite being produced on a Sixties network TV budget and having to fill 50 minutes of time rather than the shorter running times of modern network TV.
ST:TOS has this general cultural rep among non-fans for "cheesiness" and lots of goofy sci-fi tropes, e.g. the "redshirt" (go watch GALAXY QUEST for a fantastic tribute to/skewering of all of this). But it's really such *adult* writing when you revisit it.
It's fun to watch eps of ST;TOS that shouldn't work in any rational sense but still hang together by pure moxie. "The Naked Time" is flat-out bizarre and the ending 'solution' is nonsensical, but every time Riley says "Now I will render 'Kathleen'...ONE. MORE. TIME!" I collapse.
If I had to name one single STAR TREK:TOS episode as "the best" (or better yet, one for someone who only knows the show through its cultural rep to start with) it would probably be THE DOOMSDAY MACHINE, aka the one with the giant interstellar planet-destroying space Bugle.
There were a bunch of fantastic guest appearances on ST:TOS (Gary Lockwood and Sally Kellerman, Joan Collins, Ricardo Montalban, etc.), but there really hasn't been any better than William Windom as Commodore Matt Decker. It's THE CAINE MUTINY if that movie were actually good.
Anyway, it's all available on Netflix if you want to give it a shot, or relive old memories. You really can't go wrong with most episodes from the first two seasons.

Well, except for that one that ends with a misinterpreted copy of the U.S. Constitution being the payoff.
BONUS: the ST:TOS that's on Netflix is the 'remastered' series with updated SFX, primarily the outer-space stuff (but also fixing a bit of the matte-painting shots as well). It's the opposite of George Lucas: tasteful and minimal and true to the original series. Definitive.
Don't worry, the Gorn still looks like a goofy stuntman in a rubbery lizard-monster suit. They didn't do that kind of stuff in the remastered versions. Just made the spaceships shots smoother and the picture quality HD quality.
One final note: you gotta love how, on Netflix, the ST:TOS episodes are all rated TV-14, with the keywords "sex, fear." You know, because Kirk occasionally lusts after a random green-skinned alien woman.

Don't worry folks, you can show this to your 9-year-old.
Okay, a final final note: all the ST:TOS movies range from excellent to fantastic with the obvious exception of the ridiculous STAR TREK V: THE UNDISCOVERED COUNTRY (and even that has a few good moments). II through IV are one long continous plot arc, and it works very well.
The first one (THE MOTION PICTURE) is the most controversial. It is slow, and very very 'thinky'. Even the Director's Cut doesn't quite repair these flaws (also it's a rewrite of a TOS episode). But it's still full of wonder, and has the greatest sci-fi film score of all time.
I grew up as a kid with THE NEXT GENERATION before I ever saw the original series or the films, so imagine my surprise to find out TNG's theme music was actually the credits overture for THE MOTION PICTURE.
Also, Ilia's Theme (which played as an overture to the film, a la Kubrick's 2001) is just staggeringly beautiful, some of the best orchestral film soundtrack music ever recorded:
(whoops, I was referring to STAR TREK V: THE FINAL FRONTIER earlier as the one really ludicrous ST movie. The Shatner-directed one. UNDISCOVERED COUNTRY is ST VI, and it's a lovely send-off. And V still has one great line: "What does God need with a starship?")
Truth be told, ST V could have been good. Budget issues were a huge problem, and Shatner made some ridiculously bad directorial decisions (that Uhura desert dance...ugh). But the premise was solid. And man, I like that dumb "row row row your boat" scene. I do.
In conclusion: I am a huge fucking sci-fi nerd, goodnight.
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