The Ark

Derived from a Latin word for "chest" (see also the Ark of the Covenant), an ark is now best known as a vessel (usually a boat, or at least boat-shaped) in which people seek sanctuary from some great cataclysm.

This doesn't necessarily mean the people go by themselves. If they're smart, they'll bring flora and fauna with them. And if they're pressed for time or resources, they may not go bodily: The ark may simply contain the distilled knowledge and culture of the civilization, and maybe some of their DNA, in the hope that they will be reconstructed or at least remembered by someone else. In this case, the Voyager spacecraft, which possess detailed descriptions of Earth and its people, could be considered arks—except that we didn't build them under the imminent threat of our own destruction.

The concept of the ark owes most of its form to the biblical story of Noah's Ark. Most iterations of the ark today either directly identify with or at least reference the Noah story, including such elements as: The Great Flood; the ark builder being divinely inspired to prepare for the flood, and mocked by his neighbors who don't know the cataclysm is coming; the ark housing pairs of animals, implied to be all the animal species in the world, who enter and reside therein compliantly without eating each other; and, the ark builder sending a dove or other bird to search for dry land to signify that the Flood is ended.

The original Ark still crops up from time to time, usually as a Public Domain Artifact for the characters to hunt down. But new arks are just as common, and these days seem to be as likely spaceships as oceangoing vessels, with the cataclysm necessitating their construction being a supernova or similarly world-destroying event. Any structure which bears the last remnants of an alien race is likely an ark.

Compare Generation Ships and Fling a Light Into the Future.

Examples of The Ark include:

Anime and Manga

In the first season of Dragon Ball GT, Bulma just happens to have one of these tucked away beneath the Capsule Corporation in order to evacuate everyone to the new planet created by Baby with the Blackstar balls.
The Spriggan film centers on the original Ark and various organisations fighting to control it.
In Tengen Toppa Gurren Lagann, the Arc-Gurren is used to house the population of Kamina City in space when the moon is on a collision course with earth.
Doraemon: Nobita sees the vision of a flood via one of Doraemon's "peek into the future" devices, and builds an ark accordingly; turns out that was a bedwetting dream Nobita had.
Fishman Island in One Piece has a gigantic ark named 'Noah' in the slums.

Comic Books

The ship that originally brought Superman to earth, along with a database containing the history and culture of Krypton.
Crops up in XXXenophile, of all places. In "Family Reunion", salvalger Otis discovers the U.N.S.S. Rojong, the first colonization slowship from Old Terra, presumed lost in space. The bio-pods are intact and should contain all of the 'lost' animals: elephants, fireflies, anteaters, unicorns...

Film

Cats Don't Dance: The basis of the movie-within-a-movie, "Li'l Ark Angel".
Ice Age: The Meltdown: All the herds head for a giant log that would serve as a boat when the flood waters come.
Fantasia 2000 contains a retelling of the Biblical story set to Pomp and Circumstance by Sir Edward Elgar, with Donald Duck as Noah's assistant.
- And some of the animals in the ark are non-anthro ducks.
An example occurs in Battle for Terra. No points for guessing what is the name of the gigantic ship housing the humanity.
The eponymous Titan in Titan A.E., a ship holding the DNA of all known Earth organisms and the capacity to recreate the planet from scratch.
In WALL-E humanity abandons a ruined Earth in a fleet of spaceships. The Axiom is sending out probes - such as EVE - to find viable life on Earth but the "ark" takes on a mind of its own and tries to prevent a return.
El Arca
Deep Impact
Evan Almighty is basically a modern retelling of the original story of Noah, with a few twists (most notably, the flood comes not from rain but from a dam breaking).
Several of 'em can be seen in 2012. They manage to be some of the least ridiculous things in the movie (which isn't to say they aren't utterly ridiculous).
Sky Captain and The World of Tomorrow. The Big Bad Dr. Totenkopf planned for a rocket ship to take a cargo of animals to another planet. As the animals are being loaded:

Polly: My God, Joe. It's an ark.

Moonraker: Drax created his space station to hold the humans who would repopulate the Earth after the deadly spores killed everyone on the surface. When the two leads see that their space shuttle is carrying a cargo of men and women:

James Bond: The animals went in two by two.
Holly Goodhead: What do you mean by that?
James Bond: Noah's Ark. This operation.

This was also the bad guy's plot in the B-movie Theodore Rex.
Is one of the episodes in the 1936 Warner Bros. film The Green Pastures, which retold The Bible from the perspective of a poor African-American child.
In The Last Flight Of Noahs Ark, the titular plane is not an ark per se, but borrows some themes due to transporting a lot of animals and becoming a boat.
Muppets from Space opens with a dream sequence in which Gonzo is denied entrance to Noah's ark because he doesn't know what species he is. Noah gives him an umbrella.

Literature

The Bible, specifically Genesis 6-9.
Late in The Hitchhiker's Guide to the Galaxy series, the characters visit such a ship, one of three sent out from a doomed planet to colonize someplace else (which turns out, naturally, to be Earth). Subverted in that the ship turns out to be just a ruse to rid the original planet of the third of its population considered most useless (namely, middlemen).
In Stephen Baxter's Flood, great ships (of the floating and spaceborne variety) are built to escape the global flood. They are, of course, called "Arks".
The spaceship in the books composing the Deepwater Black trilogy turns out to contain the genetic structure of pretty much every animal on Earth, at which point the characters point out themselves they're on an ark. Then it turns out there were actually two arks—the other one contains all of humanity's genes.
In the YA novel Devil on My Back by Monica Hughes, the hero's Domed Hometown is called Arc One, generally assumed to be due to the curve of its roof, but the hero eventually discovers that it was originally Ark One, built so that the population could survive The End of the World as We Know It.
Timothy Findley takes a rather cynical view of the Ark in Not Wanted On The Voyage, with Dr. Noah Noyes as a despot tyrannizing his family during the Flood. The story muses on a despondent Yahweh, on the long-suffering Mrs. Noyes, on the lady devil Lucy who marries one of Noah's sons, on the innocence of the animals, and the general follies of being human amid the affairs of the gods.

Live-Action TV

Doctor Who:
- In the First Doctor serial "The Ark", the Earth is destroyed by the sun going nova, and a spaceship carrying the surviving humans, as well as samples of the animal and plant life, sets out to colonize a distant Earth-like planet.
- In the Fourth Doctor serial "The Ark in Space", the Earth is rendered temporarily uninhabitable by solar flares, and the surviving humans, as well as samples of the animal and plant life, sit it out in suspended animation on a space station. (The station commander even adopts the name "Noah"; his second-in-command tells the Doctor that they know it's not much of a joke, but under the circumstances they were taking their laughs where they could find them.)
- In the Fifth Doctor serial "Frontios", the Earth is doomed to "a catastrophic collision with the Sun", and a spaceship carrying refugees sets out to colonize a distant Earth-like planet. (The ship itself is not such a focus in this story, which begins after it ~~lands~~ crashes, resulting in the loss of most of their advanced technology.)
- In the Eleventh Doctor episode "The Beast Below", the Earth is rendered temporarily uninhabitable by solar flares, and the planet's population sets out in spaceships to find somewhere else to live until it's habitable again.
- For a non-human example, the Genesis Ark in the Tenth Doctor episode "Doomsday" turns out to be one of these, containing a refugee population of Daleks.
This happens occasionally in the Stargate Verse. And strangely, too—both Stargate SG-1 and Stargate Atlantis featured cases of civilizations trying to survive by storing just their inhabitants' minds, which, admittedly, would save a lot of resources. The ship that brings the Ancients back to Atlantis might qualify too, though it wasn't sent out with that intention.
A Filmation series in the 1970s on CBS Saturday mornings called Ark II peddled a ~~environmentalist~~ conservationist message.
The setting of the infamously-bad SF program The Starlost was the 8,000-mile-long generation ship called "Earthship ARK".

Oral Tradition, Folklore, Myths and Legends

Probably one of the oldest tropes in existence, since it was referenced in The Epic of Gilgamesh. In the story, Gilgamesh seeks out Utnapishtim, the man who survived the Great Flood by hiding inside a boat. (There's a Great Flood myth involving some kind of boat in pretty much every major civilization.)

Tabletop Games

Traveller Classic Double Adventure Horde: The unnamed inhabitants of a planet in the Alenzar system were about to be wiped out by a plague of carnivorous animals. They created spaceships with mechanisms to freeze the passengers, who would exist as Human Popsicles for the long trip to another solar system.
Warhammer Fantasy Battle has a Subversion in form of the Black Arks. Dark Elves created them to escape after a lost civil war, but now use them for waging wars and Pirate raids.

Video Games

The Yamato Ark in Ōkami brought the gods to earth after Yami sacked heaven. (Unfortunately, Yami came along...)
The main character in Terranigma is named Ark. Why? He's The Chosen One who has the power to revive the dead surface of the earth and all its inhabitants.
Ishra's Ark, a large airship of indeterminate origin, serves as one of the most memorable levels of Klonoa 2
The Halo universe contains a Forerunner megastructure (no points for guessing what its name is) which is approximately 262,144 lightyears from the centre of the Milky Way Galaxy, well out of range of the Halo superweapons that kill everything within 25,000 light years of them. It too is to escape "The Flood", represented here by ancient alien zombies.
- The Ark is not only the ultimate Forerunner defence in that it serves as, well, an ark, but also because it provides a remote detonator for any and all of the Halo rings.
- Halo also has the Shield Worlds. 343 Guilty Spark hypothesized that the Ark would be in one, but was proven wrong.
The entirety of the game Brink is set on a Deconstructed Utopia floating along the surface of a flooded Earth (no points for guessing what this place is called, either). It was designed as a peaceful and fully sustainable city easily capable of being home to 5,000 people; however, everything gradually fell apart as the population grew to about 10 times as much, and the city is now on the brink of a civil war.
Doom 2: After the demon hordes of Hell invade Earth, the remaining human population is loaded onto space ships that will carry them to safety.
World of Warcraft has both the Exodar and the Oshu'gun.
The Space Colony ARK from a handful of Sonic the Hedgehog titles is apparently a long-term habitable structure, but has remained inactive for the majority of the time we see it, staying in orbit over the planet.
The planet Ilos in Mass Effect. Unfortunately, the Reaper harvest took much longer than their power supply could keep the Protheans on Ilos in stasis for, and it instead became a massive tomb for all but a handful.
In Honkai Impact 3rd, one of the plans the Previous Era had for surviving the Honkai-induced apocalypse was Project Ark, the sending out of a spaceship to find a inhabitable world to evacuate to. It was never heard from again.

Western Animation

Yogi Bear and his pals manned a flying version in Yogi's Ark Lark, the Pilot Movie for Yogi's Gang, a cartoon that premiered on ABC in 1972 and justifies the trope The Dark Age of Animation.
The Autobots in Transformers traditionally come to Earth in a ship called the Ark, usually relating to how the planet becomes their second home for some time for one reason or another.