Signed Languages are languages which primarily function non-verbally through visual signals, generally invented for the use of the deaf to communicate. As the name indicates, the primary means of communication is generally signs made with the hands in front of the body. However, most sign languages include facial expressions and some, such as Japanese Sign Language, include mouthing as part of their mechanics. It is important to recognize that while almost every community with a spoken language also has a signed language, the signed language used is related more to the geographical region than to the spoken language. For example, English is the primary language of the United States, Canada, UK, Australia and New Zealand, but the US and Canada use ASL, the UK uses BSL, and Australia and New Zealand use AUSLAN and NZSL respectively - all related languages but with distinct signs and grammar.
One important aspect of Signed Languages is that they are, as a rule, fully formed languages with their own grammar and words. They are not pantomime nor do they necessarily follow the grammar of the spoken language of the region. Some signs are iconic, or resemble what they speak of, much like how some spoken words are onomatopoeic, but most signs are abstractions of iconic signs or completely original. The grammar itself frequently differs greatly in part due to the spatial aspects of signs and the ability to convey information non-sequentially. For example, within ASL, it is common to establish specific people in a conversation at spatial locations and later use signs moving from location to the other rather than having to reestablish identities or use pronouns. Similarly, since both hands and the face can be used, multiple pieces of information can be encoded into a single sign. For example, a sentence like "I drove from Jane to John and I enjoyed it" can be conveyed in a single sign if Jane and John have already been previously established in the conversation.
It is worth noting that Signed Language, while non-verbal, is not necessarily quiet. Even deaf users typically make sounds while signing and it is not infrequent for a very low-pitched grunt to be used to catch someone's attention via the vibrations.
Signed Language has nothing to do with the trope of Talking With Signs which involves characters communicating via written signs. It is related to Hand Signals, which range from pantomime to a reduced vocabulary, sometimes with a sparse grammar. Especially within fantasy works, it is not uncommon to have races or nations where Hand Signals have evolved into a Signed Language, typically to provide a method to communicate in secrecy.
The following works involve Signed Language as a significant aspect of the plot:
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Anime and Manga
One of Gangsta's main characters is a deaf Twilight named Nicolas Brown who primarily communicates with his Heterosexual Life Partner Worick and friends using sign language. Note that Nic can speak, he just hates to do so because its tiring and comes off... not quite right.
The Bronze AgeTeen Titans version of Jericho communicated only in sign language (his vocal cords had been cut by his father's enemies). If he spoke at all, it was because he was possessing someone. Marv Wolfman also prohibited the use of thought bubbles when writing him in the comic, leading George Perez to get creative when displaying hand gestures. This trait is carried over into his animated appearances.
All of the undersea settlers in Dark Life know sign language, because the liquid oxygen substitute they use when diving keeps them from talking out loud.
In What Women Want, there's a moment when Nick Marshall believes he has got ridden of hearing women's thoughts. Until he sees two deaf women using signed language and he can hear what they're saying/thinking.
The Crucible, a 2011 Korean film based on the true story of a sex abuse scandal at a school for the hearing-impaired, naturally has a great deal of subtitled KSL (Korean Sign Language).
With the exception of one line in English, Marlee Matlin gives her Oscar-winning performance in Children of a Lesser God entirely in American Sign Language. It isn't subtitled; instead, William Hurt's character interprets to himself, out loud for the audience's benefit. When he doesn't, Matlin's expressiveness speaks for itself.
The Drow hand code in R.A. Salvatore's novels (the ones with Drizzt Do'Urden). Drow elves are all taught a language composed entirely of hand signals. Any two Drow can communicate in this way in complete silence as long as they can see one another. And in complete darkness, because of their infravision.
Some surface Elves in Forgotten Realms, according to Return of the Archwizards, have "finger talk". As opposed to at least part-"hieroglyphical" Drow signs, it's alphabetical language and at least to some degree useable with human hands.
The Finder's Stone trilogy mentioned thieves' hand cant. Saurials can't speak aloud, so it came in handy, teaching a paladin to understand it.
The Clan in the Earth's Children communicate primarily by sign language and Body Language, using vocal noises only for names (based on a now-disproved theory that Neandertals had less-capable vocal cords than do Homo sapiens and could not have supported a fully-verbal language). Visual miscommunication is Played for Drama several times in Clan of the Cave Bear, with The Resenter always turning away before someone compliments him.
The Drasnian secret language, of the Belgariad, by David Eddings. All Drasnians involved in the intelligence community (which apparently means "all of them") are taught a language similar to the Drow version above. On more than one occasion, two such speakers converse verbally about something unimportant while having a completely separate discussion with their hands. The language is specific enough that a speaker can gesture with a recognizably outlandish "accent": when Garion first learns it, Silk notes that his initial use of it is a bit off due to learning it in a cold environment (with finger joints frozen).
Verghastite, and later some Tanith, members of Gaunt's Ghosts that have been deafened by artillery fire communicate with a form of sign language.
In one of the Xanth stories, two characters learn American Sign Language as a way to communicate because a Xanthian girl cannot speak any known Mundane language. They later talk to a deaf man on the bus, because he saw them signing and thought they were deaf, initiating a conversation in American Sign Language.
In Dune multiple characters use hand signals to give orders to their subordinates. In fact, there are entire sign languages developed separately by the Atreides and the Harkonnens that allow them to communicate irrelevant information verbally and important stuff, with their hands, making sure that even if they are overheard, the enemy won't learn anything.
In Mirror Friend Mirror Foe, a ninja family is not only trained in that... They can communicate that way while having a verbal conversation on a totally unrelated matter.
The giant raccoons in Architect of Sleep use sign language to communicate due to having never developed a complicated enough vocal apparatus to support a spoken language. They do punctuate their signs with trills and chirps, though.
The Isitri from the Star Trek novel "Troublesome Minds" by Dave Galanter. They communicate by a mixture of telepathy and sign language, and have no spoken language (they have poor hearing, with many being deaf, and their throats and mouths aren't configured for verbal speech). The sign language is used by the few non-telepathic Isitri, and by all Isitri to communicate with aliens.
The telepathic treecats in Honor Harrington communicate with humans via sign language derived from ASL (modified for the treecat's Four-Fingered Hands), although they can understand spoken word fine.
The crypt workers in the People's Palace. Darken Rahl had their tongues cut out so they wouldn't speak ill of his father, so they developed one of these. Only their boss understands them at all, and poorly at that. He still has his tongue, primarily for this reason. Cara begins to follow what they mean after a length "conversation". When they get their tongues magically restored, they never stop talking. It has less Unfortunate Implications than one might imagine, because these people were all mute against their will, and had no support structure like actual deaf and mute communities, so they would be less inclined to stay mute.
The Heralds of the Heralds of Valdemar series not only have a hand-based signed language, but also a form of "Hand Signals" involving apparently playing footsie under the table for when they wish to communicate in secret.
In the Sime Gen world, Householders at a Gen auction will hold hands, the Sime wrapping tentacles around their hands as well, and communicate via subtle pressures and shifts in contact. (There's also a great deal of non-verbal information communicated via manipulating nagers.)
The second book in A.C. Crispin's StarBridge series deals with a species of crane-like aliens who communicate primarily in sign; vocalization is used only as emphasis or warning...and their voices are so loud and piercing that they do physical damage to fragile human eardrums. The protagonist is a deaf cultural interpreter who is brought in to live with them, meaning that most of the book's dialogue is in sign.
In PartnerShip, the second book of the The Ship Who ... series, Blaize learns that the native Anglians on the planet he's tending can communicate in sign language, and are reasonably intelligent once they learn they can communicate with humans.
Live Action TV
The old CBS TV show Beauty and the Beast had a deaf character who had grown up in the tunnels in "An Impossible Silence" and "Sticks and Stones" who communicated through ASL. The second episode was groundbreaking in that there were several scenes where deaf characters communicated in on-screen silence, with no voiceover or even background music, something the deaf actors involved fought hard for, not wanting someone else's voice to overshadow their own "voices."
CHiPs: they meet a deaf woman and learn sign language just in time to use it during a chase scene when they were far away from each other and there was construction noise nearby; then they never used it again.
Shows up a lot in NCIS as Abby and Gibbs both know ASL. Actually turns up again NCIS Los Angeles when Abby is kidnapped and uses ASL to warn the local team about the room she's in.
Sesame Street included Linda, a deaf woman who communicated entirely in American Sign Language during the course of the series.
The BBC has a magazine show for deaf people called See Hear, with presenters and interviewees using British Sign Language throughout. Until fairly recently the BBC would also repeat various prime-time shows in the early hours of the morning with an in-vision sign language interpreter for use as a teaching aid.
A children's show that aired in the late 90s/early 2000s called What's Your Sign? had one hearing host and one deaf host, and everything spoken was also said in ASL, to help teach kids sign language.
In the The Saga Of Tuck, Tuck is fluent in ASL because he had communication problems when he was a child, so his family and Mike learned it as well. Comes very handly to secretly talk in plain sight without listeners and to befriend a deaf girl in Seasons.