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Sex & Nudity
Sex: Very occasional super light references at most (less than once per season - if at all.) What little happens is off-screen. One of the Doctor's companions, Jack Harkness, is "omnisexual", i.e. he'll flirt with / hit on anyone: male, female, or other; human or alien. He's not pushy about it.
One of the Doctor's Season-10 companions, Bill Potts, is lesbian: she's attracted to and dates women, and has no romantic interest in men. Nothing is shown beyond those points and no issue is made of the matter.
Nudity: Some by implication; little shown. In one Season-1 episode, "Bad Wolf", Harkness is zapped by a "defabricator" ray on a far-future TV fashion show, which disintegrates all of his clothing. He is shown from behind down to the top of the buttocks, and full-body from the front with the defabricator device blocking view of his genitals.
Violence & Gore
Not a whole lot of violence, and what is seen in terms of death is quick and bloodless.
Mild to moderate language. Many uses of "damn" and "hell," some uses of "bloody"
Series 10 has some uses of "arse"
Series 10 Episode 6, Extremis, has quite a few uses of both "arse" and "ass".
2 uses of "bitch" 1 use of "goddamnit" and "cock"
Also there is an unfinished use of "shit" in Series 10 Episode 3, Bill says: No sh-
Then it cuts to a new scene.
Alcohol, Drugs & Smoking
Alcohol consumption and smoking due to some historic set stories. Science fiction drugs are featured.
Frightening & Intense Scenes
The episode Extremis has many suicide references, and one off screen but audible suicide.
Season 8+ episodes are rated higher, realising the audience is mostly long time followers (i.e. getting older) more adult approach has been taken. (As mentioned above including a few more deaths, but no where near the gore of US TV... deaths are instant & bloodless, sex is still barely, if at all there, zero nudity).
The previous Doctor Who (1963) series originated the "Behind the sofa" trope of children watching scary scenes from a place in the room where they felt safe. The current series can be more intense than the previous series, with more realistic special effects and more time devoted to building atmosphere.
BBFC: 1 episode U, 111 episodes PG, 19 episodes 12.
In the UK, Series 1,2,4,6,7,8,9 and the 2008-10 specials are rated 12.
Series 3,5 and the 50th anniversary specials are rated PG.
Over all it would be rated 12: Contains moderate threat, horror, violence, scary scenes, infrequent moderate language, sex references and gory moments.
In the USA, every episode is rated TV-PG, however some should be rated TV-14.