Sword Limbo

A common maneuver in a Sword Fight: one character will dodge a sword swipe by bending over backward as the opponent's blade whips horizontally over their body.

Often filmed from above and in Slow Motion to boot, ala Bullet Time. Sometimes the shot will also employ an extreme zoom to show the sword cutting through a loose strand of hair. Ooh, that Could Have Been Messy.

Not at all about swords that disappear into nowhere.

Examples of Sword Limbo include:

Anime and Manga

Hei of Darker than Black dodges a stream of blood (that Portal Cuts on contact) in this manner.
Rock Lee pulls this off while drunk during his fight with Kimimaro in Naruto.
Mugen during his fight with Sara in Samurai Champloo.
- Mugen also pulls this off in his first fight with Ukon/Shoryuu.
Revy from Black Lagoon does this to duck Shenhua's throwing knives so that they can kill the bad guys they were meant for instead of her.
Fakir pulls one off during the finale of the first chapter of Princess Tutu when attacked by sword-wielding ravens.
Kaku from One Piece does this during his fight with Zoro.
- And Brook in his fight with Ryuuma, complete with the Close-Call Haircut (kind of inevitable given the size of Brook's Funny Afro).
The titular character of Ranma ½ does this several times, absentmindedly (he was more interested in looking at some photos,) during his first real duel with the bokuto-wielding Tatewaki Kuno. The latter is extremely irked at Ranma's nonchalant dodging, claiming Ranma wasn't taking him seriously (which was absolutely right.)

Live Action TV

Happens with Sam Carter in the "Emancipation" episode of Stargate SG-1, albeit with a knife.
In the Legend of the Seeker, Kahlan and Cara do this all the goddamn time. It turns out that when you have magic and you're fighting mooks with swords and all you're armed with is daggers or an Agiel, this is the proper way to not be exsanguinated. For those who don't know, an Agiel is a torture device that looks like a cheap dildo. We have no idea either, but it certainly doesn't have reach on swords or polearms.
- Sisters of the Light and Sisters of the Dark occasionally do this, often against Richard.

Film

The Bride in Kill Bill vol. 1, in her fight with Gogo and later the Crazy 88.
The Film of the Book The Lion, The Witch And The Wardrobe: Peter dodged a stab from the White Witch in this way.
Popularized by Neo's bullet dodging sequence in The Matrix. The sheer number of times this particular variation has been parodied itself is enough to warrant its own article. Why this trope wasn't titled Neo Limbo is a mystery in itself.
- A melee example occurs during the "Burly Brawl" battle in Reloaded, where a ring of Smiths all bend out of the way when Neo swings at them. (From the Knox voiceover: "Limbo!")
- It's also in The Animatrix.
In the first Spider-Man film, Peter Parker dodges a punch in this manner. He later avoids Green Goblin's razor blades with it.
Similarly in Daredevil, during their first encounter, Daredevil is able to successfully dodge razors thrown by Bullseye with this, which naturally pisses the hell out of him.

Music Videos

OK GO's "A Million Ways" throws one of these into its choreography, complete with impersonated bullet time.

Professional Wrestling

Though they're dodging fists instead of swords, WWE Divas Trish Stratus and Melina both use this style of dodge frequently, and a few male superstars (Rey Mysterio, Jr. and Jeff Hardy most notably) have used versions of it once in a while.

Video Games

This is a standard dodge animation of trolls in Neverwinter Nights, both more justified than normal in that they can bend much farther backwards than humans, and less justified in that they may do this even if you're using a downwards slash that you can clearly see going right through them.
Warhammer 40k: Dawn of War 2 opening. Performed by an Eldar Banshee to evade a fuckhueg chainsword.
Pulled of by Thane if he's still alive to fight Kai Leng, in a particuarly awesome confrontation between two assassins. Thane can't pull it off the second time, however.

Western Animation

In Futurama: Bender's Big Score!, Hermes shows off his skills by literally limbo dancing under a sword. It's subverted when he gets his head severed by the other sword that was mounted on one of those sword/shield/wall mount thingies.
Avatar: The Last Airbender has used this twice so far:
- Zuko to Jet in "City of Walls and Secrets"—In a "Making Of" special, Bryan Konietzko and Sifu Kisu were doing a motion-reference for the Sword Fight between Jet and Zuko (Bryan was Jet, Kisu was Zuko) and they did this part in slow-mo (as in they intentionally did it very slow to show of the rate of time; Bryan even had a blade of grass in his mouth).
- Sokka in "Sokka's Master", as shown above.

Real Life

While this can be done in real fencing or other styles of sword combat, it's generally not advisable. This is because it pretty much involves leaving your entire upper body open to attack if it doesn't work. That, coupled with the fact that it's pretty easy to predict and thus compensate for, gives you a move that rarely works and very inefficient...but looks awesome.