Cousin to invisibility, intangibility refers to a state where a (usually visible) entity cannot interact physically with other matter. Everything just phases through the affected object.
Those questions you have about how this could possibly work? Don't ask them.
A truly Intangible Man would sink through the ground and yo-yo around the Earth's core at the whim of gravity forever. How they can generate enough friction to move around, let alone grip things if the writer is feeling charitable, is also likely to be ignored (occasionally these problems are avoided by also giving the character the ability to fly). Asphyxiation is generally not an issue, potential breathing problems are typically ignored outright. If they're stuck in an intangible state for a while, starving to death usually isn't brought up either. If they're invisible as well as intangible, don't expect anyone to know that they shouldn't be able to see (if the light passes straight through them, it can't form an image on their retinas). Again, don't ask why.
Although the Intangible Man will pass right through walls, chairs, tables and such, he rarely has any problem properly interacting with floors, stairs or any other structures people are meant to walk upon. He may even be able to ride vehicles, with the odds of success rapidly increasing with the size of the vehicle. Anything big enough to walk around inside is quite likely, a ship or spaceship is more or less guaranteed.
Note that once we've seen someone walk through the character, to establish the parameters of their intangibility, people will go out of their way not to walk through them too often, as this is an expensive special effect. This is usually justified as the experience being intensely creepy.
For a mentor, intangibility is an opportunity. Being a Spirit Advisor or Virtual Ghost lets them accompany the hero into dangerous places without ever being at personal risk.
The ability to render oneself intangible at will occasionally turns up as a superpower. Sometimes the power can work selectively on the user's body (allowing him to voluntarily interact with objects while intangible) or extend further from the user's body (allowing him to make other objects and people intangible) Heroes use this ability to save their teammates from danger. Anti-heroes and villains use it to put their hands through people's chests and squeeze their hearts. It is frequently a given that while no one else can touch them, two characters separately rendered intangible by the same process will have no problem interacting with each other.
See also Projected Man, Astral Projection and Super Smoke.
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In this Pepsi commercial, two nerds make themselves intangible and get into all sorts of hijinks. At the very end, they realise they can't even pick up their can of Pepsi anymore.
Anime and Manga
Nancy Makuhari and her son Junior from Read or Die have this ability. It's explicitly stated that they can't breathe while phasing.
Kurau from Phantom Memory Kurau has this power as well - it's one of her more rarely used powers, as it tends to get her noticed more than superstrength or even flight.
In NarutoTobi is able to become intangible by using the same space/time manipulation he uses to teleport. The primary weaknesses are that he cannot attack while intangible, he cannot be intangible while teleporting himself or others (and the more he tries to teleport the longer it takes), and he can only stay intangible for five minutes in a row. He found a novel way to get over the first one: by attaching a chain to arm manacles he runs toward people, phases through them when his body would touch them, and then turns solid again just in time for the chain to hit the target.
Ryoko of Tenchi Muyo! has been shown to walk through barriers and walls unless they are specifically shielded to prevent her from doing this.
Sasami even comments on this at one point in an episode with a little ghost girl, who asks if her phasing-through things disturbs Sasami at all.
A manga series called Tokyo ESP by Hajime Segawa (who createdGa-Rei) features a protagonist, Rinka, who gains this power after a freaky incident involving otherworldly fish and arctic mammals.
Perrier La Mer of Amuri in Star Ocean has this power (called Infiltration in the show) but cannot fully control it, so she's always passing through other people and solid objects. Her parents took advantage of this by making her the victim of dangerous stage performances like knife-throwing acts and passing giant buzz-saws through her body. None of it caused any physical harm, but she was still deeply traumatized.
Some users of Logia Devil Fruit in One Piece can do this by transforming into their particular power's element (Ace, God Enel, Admiral Kizaru), becoming immune to most attack. The only way around this is to attack them with an element-specific weakness, or be able to use a form of Haki called the "Color of Armaments", which allows its user to bypass Devil Fruit based defenses.
This is also a part of Perona's Devil Fruit power. The Hollow-Hollow Fruit makes Perona a "Ghost Woman", allowing her to generate and control ghosts that drain positive emotions from people. She can perform Astral Projection, leaving her physical body behind while enabling her own spirit to move freely. She can't make herself tangible without returning to her body, so while she scares Usopp with the threat of 'heart squeezing' mentioned elsewhere on this page, she can't actually do it.
Evil Weapon Anubis from Jojos Bizarre Adventure is an Egyptian sword who's blade can pass through objects to cut an object behind it, as demonstrated soon after its first appearance, when it slices a cow and a man behind it, but only cuts the man.
X-Men: This is Shadowcat's mutant ability. Her ability to not fall through the floor, while not explained well, has nothing to do with the actual, physical floor (which she can pass through if she chooses) and, as such, can be used without a floor. She often walks on air by visualizing an invisible staircase. Later works have treated it as slow flight. This tends not to show up in adaptations.
Joss Whedon created a similar situation in Astonishing X-Men, where Cassandra Nova mindscrewed with Shadowcat and caused her to phase deep into the Earth.
Whedon also had Kitty accidentally lose control enough to phase through the floor when she and Peter Rasputin first did the deed, in an amusing display of Power Perversion Potential.
Also from X-Men, the villain Shinobi Shaw has the same power. Being a villain, he uses the 'heart squeezing' as his trademark killing move. He also openly admits it can be used for peeping on girls (not that he needs it with his cash)
Negative Man from the Doom Patrol can leave his body as an energy-form that can control its tangibility.
In The DCU, all natives of the planet Bgztl have voluntary intangibility powers. This is the home planet of Phantom Girl, from the Legion Of Super-Heroes.
Well, in the Threeboot version it's explained as Bgztl being a planet in another dimension which exists in the same space as Earth and her shifting her mass between our dimension and that one as a unique power. She's visible in both worlds (which gets awkward in the one she's not paying attention to) and is solid and can sense in only one at a time. This solves the floor problem (and causes a new one when someone has to go to the core of a planet) when you realize that she can't avoid the planet by shifting to a dimension with an identical planet in the same place.
Iron Man enemy and Thunderbolts member The Ghost is a technological industrial terrorist.
Also Iron Man has an industrial spy fellow with this as a secondary ability.
In Dark Horse Comics' Ghost, the titular character could make herself intangible (and fly, thus avoiding the floor problem). She had a particularly nasty attack where she would reach into an opponent's chest while intangible, grab his heart, and then become corporeal again. (Why her hand wasn't hurt in the process is not explained.)
The Martian Manhunter has this power, also having flight powers so he doesn't have to worry about falling through the floor.
Marvel Comics' The Vision was a flying android who could shift his molecular density, from intangible to denser-than-diamond. He made frequent use of this power to reach into other beings and dodge attacks.
This includes phasing while someone is kicking at him... like Batman.
Moonstone in Thunderbolts (later the Dark Avengers version of Ms. Marvel) can turn intangible. She has outright flight to deal with the falling-through-the-ground aspect.
Hero Hotline featured Fred, the invisible, intangible man. The problem being that he can't interact with his surroundings at all. It's speculated that Fred isn't real and is just a ploy by super-ventriloquist Voice-Over to pick up two paychecks, but V/O talks to Fred when nobody else is around, so maybe not.
The Daredevil villain Death-Stalker was trapped in a dimension partially connected to Earth's dimension, and while naturally invisible and immaterial, could become visible and intangible, invisible and tangible, or visible and tangible for a few hours at a time. He also wore gloves that used microwaves to give him a "death grip". He died when he shifted to full tangibility to grasp Daredevil with the gloves, but was halfway through a tombstone; the trauma and shock killed him instantly.
Blue Fire, from Wham Comics, can become intangible and is covered in blue flames. It was explained that his intangibility was related to force he or another used, the gentler he touched something it could be solid, but a punch would pass through.
The villain Hood, from Amazing Man Comics, has the powers of intangibility and teleportation. The Amazing Man himself could become intangible when he turned into the Green Mist.
The Duke of Darkness, from Triple Threat Comics, uses this power to battle villains such as Mr. Slumber.
Sergeant Spook, from Blue Bolt, has this power as well as invisibility.
Guardians of the Galaxy has minor villain Brahl, who can turn intangible at will to keep from being hurt. Subverted when he's decked anyway by a hero who's also intangible.
New 52 introduces a new Superman villain named Anguish, a woman with this power. Anguish explains that her power comes from altering her body's density, meaning she can also increase it to the point that she's strong and fast enough to kick Superman around. She complains that her power activates automatically, which is useful against sneak-attacks, but not so great when she actually wants someone to touch her. Superman only manages to hit her once, by breaking the ground that she's standing on. In the end, Superman resorts to fixing and returning her beloved mother's locket, at which point she leaves.
The Silver Surfer can do this due to his manipulation of molecules but doesn't do it often. His movie counterpart did this quite a bit, particularly in his introductory scene where he phases through buildings, cars, the Human Torch, and even his own board.
Sublime of DV8 has control over her density — she can walk through walls or be tougher than diamonds. She mostly goes for the latter choice, being a very talented and skilled hand-to-hand fighter and all.
In Whip And Wing, a supporting character, a unicorn named Silver Trowel, has an enchanted necklace that allows the wearer to pass through walls and avoid injury. It helps save himself and Indiana Jones from a few tight spots. Much more problematic when Trowel pulls a Face Heel Turn, and Indiana can't catch him. Indy finally kills him by grabbing the necklace, pushing Trowel into a wall, and removing it. This forces Trowel to materialize with his upper body inside the stone, and he suffocates to death.
The monster in 4DMan was a scientist who acquired this ability. Using this power consumed his life force at an accelerated rate and he had to replenish it by phasing through other living human beings.
The movie Ghost goes out of its way to show how much effort it takes newly dead Patrick Swayze to move a penny, but he never has the floor problem, nor does he have a problem riding in elevators.
This and Ghost Dad can mostly handwave the floor problem away because they are actually ghosts, and presumably susceptible to different physics/laws. Or whatever...
The Bill Cosby movie Ghost Dad has Bill as a ghost who can only interact with the real world if he "concentrates". This results in a scene where his concentration lasts enough for him to pour himself a drink, but when he tries drinking it it goes right though.
The Twins in The Matrix: Reloaded can turn this on and off at will, and use it selectively in fights to dodge attacks. It has the added bonus of healing wounds - several gunshots to the arm can be undone simply by turning intangible and then reverting. Morpheus blowing up their car with them inside it had more of an effect, although whether they survived is left ambiguous.
In Labyrinth, Jareth phases through Sarah during "Within You".
In the Night Watch universe, descending through enough layers of Twilight will turn the Other intangible. The "falling through the floor" problem, however, is not ignored at all - in fact, the very trick is used to climb a tower through the floors in the third book.
In the Wild Cards universe, Modular Man turns intangible by shifting his mass partly out of this dimension. He also defies gravity by the same method. Both powers are lost when his inventor turns Joker and Mod Man can no longer be repaired.
The Astronomer also had this as one of his powers. It backfired when he tried to use another one of his powers, a mental shield, to protect from another character's literal Death Glare... and since he could only use one power at a time, and was halfway through a wall...
Wraith is another character with intangibility who decides to make a living as a Classy Cat Burglar. However, as she's got maybe a ten pound limit to objects she can carry while phased, she performs her heists in a mask and bikini and focuses on paper goods (such as high-currency bills, bonds, etc.).
Mackie Messer, aka "Mack the Knife," could vibrate himself at such frequencies as to walk through walls. He didn't seem to have problems with floors.
In Chris Wooding's novel Storm Thief, there are creatures called revenants, that are totally intangible, and if they touch you you die. They are also invisible, and there are basically only three ways to kill them. The first is to use a specialized weapon called an Aether Cannon, the second is to have a unique golem touch them, and absorb them into himself, and the third involves total destruction of the Chaos Engine, an ancient artifact, and a massive one at that that is next to impossible to destroy.
This condition was inflicted on an unwitting thief by a cursed amulet in a short story from Tales of Ravenloft. Floors weren't a problem, but eventually the poor guy became so intangible that he completely faded out of the world.
In The Skinjacker Trilogy by Neal Shusterman, the Afterlights (ghosts) are like this. They can only stand on ground that has "moved on". The requirement to move on is being loved and cherished by many people. If they stand on 'living' ground too long, they will sink into the center of the earth.
There's a German SF short story out there in which a scientist invents a machine that can turn people intangible for a time and gets blackmailed by a thief who wants to help himself to that power. The thief hasn't thought things through, though, and so when the scientist does grant his wish, he promptly falls through the ground (and presumably his death once the effect wears off again), a fate the scientist only averted during an earlier self-test by keeping his own feet unchanged and solid. In the end he decides that his device is actually rather useless, save possibly as a novel method of execution.
In Leven Thumps, this is one of the twelve gifts you can receive when you arrive in Foo.
Death from Discworld is more than capable of walking through walls. Rather than becoming intangible, he's simply so real that the rest of the universe may as well be made of cobwebs.
The Saga Of The Noble Dead has the undead wraith Sau'ilahk. He is a powerful sorcerer and can turn solid for brief periods to move physical objects, but is most dangerous in his intangible state: any living thing that passes through him is rapidly drained of life.
One of the characters in Orson Scott Card's Pathfinder novel can do this. Although it's not really phasing, it is actually a limited form of Time Travel. The character continuously jumps forward in time by seconds/microseconds. And it is painful, and potentially even deadly to actually phase through people/objects.
In the short story "What Doctor Gottlieb Saw" by Ian Tregillis a Stupid Jetpack Hitler experiment gone wrong causes the subject to fall into the centre of the Earth. His successor in Bitter Seeds has learned to focus his powers to prevent this happening, but he can still only remain intangible as long as his breath of air lasts.
The spirit haunting a space station in Galaxy Of Fear: Ghost of the Jedi glides about and passes right through walls and any other solid object, though he can affect Tash. He feels helpless and like a failure, but makes a decent Spirit Advisor to her.
Live Action TV
Star Trek: The Next Generation: In "The Next Phase", Geordi and Ro Laren were thus afflicted by an experimental Romulan cloaking device which also served as a "phasing device". It made them invisible and intangible, making them think they were dead ghosts.
Later it is revealed that the UFP also experimented with such a cloak. It can render the ship intangible. but a treaty forbade it and the system was Gone Horribly Wrong.
It is heavily implied that transphase torpedoes work this way to get past defenses.
Stargate SG-1 did this at least three times: A race of highly advanced humans had developed this technology; Daniel was once thus afflicted by a mysterious alien crystal skull; and SG-1 itself was placed in an intangible state by a security device belonging to Merlin (secretly an advanced alien).
There was also an episode where Carter, trapped in an Asgard research facility, used the holographic comm system to help O'Neill and Teal'c rescue Thor from a Ha'Tak in orbit. Since the Asgard lack Hard Light technology she was able to distract guards and scout ahead, but proved less useful when they needed someone to rewire doors.
Angel: Afflicted Spike after he was brought Back from the Dead. He, however, was a ghost or some other sort of psychic projection, and didn't have to worry about falling through the floor because he was essentially imagining himself wherever he liked.
Also Sahjhan from Season 3. For most of his appearances, anyway.
And, of course, the First Evil, whatever it manifests as. It's a plot point in the middle of the last season that Giles may be the First because he doesn't actually manipulate anything.
One episode of The X-Files had an escaped convict who had this as a power, as well as the ability to reduce anything he passed through to ash. It's demonstrated with some Technobabble that it's electricity-based and doesn't work against insulating materials, and he's eventually done in when he tries to phase through a speeding car and gets fatally smacked by the windshield.
D. L. Hawkins, from Heroes, has this as his special power. He can use it selectively, allowing him to touch one object while phasing through another, thus providing a handy rationale for the "floor problem". He can also use it in a lethal fashion, such as when he literally rips Linderman's brains out.
In the short-lived B Sci Fi show Cleopatra 2525, the 'falling through the floor' problem was averted; the Big Bad revealed it had taken years of research to solve.
The mentally-projected hologram of Al on Quantum Leap appeared to have this power, but he wasn't really "there." He was able to give the impression of intangibility to a man who was able to see him in the episode A Little Miracle.
A natural ability of Dessarians in Tracker was the ability to phase and walk through solid objects.
The Eureka episode Phased and Confused had this as the intersection of its A and B plots. The mysteriously solid floors became very noticeable when a phased character noisily runs down a short staircase, immediately after not being able to touch things was an obstacle.
MarillionThe Invisible Man. Despite the title, the lyric describes an intangible man.
Similarly, one of the powers potentially available for a Rahkshi (their creators, the Makuta have it by default) is Density Control. This can be used not only for phasing through walls and attacks, but for floating into the air and then dropping with the force of a meteor. Makuta Krika frequently made use of this power after a mutation caused him to lose most of his other abilities. It also lead to his death. When forced out of control, his power turned him so intangible that he simply ceased to exist.
Wearers of the Kakama Nuva, the enhanced version of the Mask of Speed, can pass through objects and people by vibrating their own molecules very fast.
In Mortasheen, the Xenogog has this power, though it can solidify the tips of its needlelike arms to perform "a surgically perfect lobotomy with a single touch"
In the collectible card game Magic: The Gathering, there are entire races of intangible people (the Dauthi, Soltari, and Thalakos). Their homeplane got caught between Dominaria and Rath and ripped to shreds, leaving the inhabitants as Intangible Men who can pop in and out of reality at will.
Necron Wraiths do this at will in Warhammer 40000. The same technology may play a role in the effectiveness of Necron melee weapons against power armor.
As do the C'tan, the masters of the Necrons.
In Deadlands: Reloaded, Harrowed get intangibility as an Edge; a power they can turn on an off at will.
Dungeons & Dragons has the "incorporeal" subtype, which cleared up a LOT of confusion about intangible beings such as ghosts or 2E Wraithform spellcasters when 3E introduced it to the game.
4th Edition splits this in two, with the Insubstantial resistance ability (half damage from all attacks) and Phasing movement ability (the ability to move through solid objects and difficult terrain without penalty, though you still have to occupy an open square when your movement is finished). Some creatures can be Insubstantial without having Phasing and vice versa, though most creatures with one also have the other.
Also the Desolidification power in the Hero System. Which works a bit asymmetrically — characters with it can't affect the physical world at all while it's in use unless they buy their relevant powers (such as strength) with a rather expensive advantage, but need to define some reasonably common way(s) in which the physical world can still affect them. They also can walk on solid ground with no problems, but don't automatically get to fly unless they buy that power separately, and may in fact fall through the floor if they don't stop themselves at the price of taking the usual damage from the 'impact'.
The fourth level of the Insubstantial power in 2E Mutants & Masterminds grants this power and handwaves the breathing and being able to stand on the floor, although you still face a breathing problem when within solid objects. It's also required that some reasonably common material or effect counters your insubstantiality for game balance purposes.
Akuma's float-on-one-knee trick from Street Fighter also turns him intangible so he can zoom around untouchable. If you ever see him doing this trick while tangible though, hit him or get out of the way because you're a second away from the Raging Demon attack.
Bow's "Out of Sight" ability in Paper Mario turns the player intangible and invisible, but you can't move while using it.
Boos can turn intangible in Super Mario 64. Mario can do this himself with the Vanish Cap. (Should have been called Intangible Cap.)
Final Fantasy VI famously has Intangirs, a kind of untouchable bonus mook, in addition to regular intangibility spells.
The DS-exclusive Violet Wisp from Sonic Colors turns Sonic into an intangible hedgehog, able to pass through walls.
Lusternia has Zenos, a particularly creepy example: basically, he's an omnivorous sentient fog. Tales abound of him enveloping whole forests and cities, devouring the inhabitants and leaving only charred wasteland in his wake.
In Dragon Age II, Fenris has the ability to become partly intangible as a result of having his body forcibly laced with lyrium. Exactly how intangible he can become is unclear; in gameplay terms, his "Lyrium Ghost" mode makes him more resistant (but not immune) to damage, and aside from that he mostly uses the ability to crush people's hearts in their chests.
Tales Of Monkey Island: Subverted at first in Chapter 5, when Guybrush can still touch and pick up the items as a Ghost Pirate in the Crossroads of the afterlife; but when he returns to the living world, it becomes Double Subverted when he tries touching the Cursed Cutlass of Kaflu, only to find that he can't touch it (or any other item) or even pick it up while in the living world (he needs to get back in his own body in order to do that).
Devil May Cry 3 has The Fallen, angels with the faces of demons who fly through walls, floors and ceilings as easily as they do empty air.
Geist from the webcomic Heist is a rare case of simultaneous Intangibility and Invisibility. While he doesn't have to worry about falling through floors, there is a two-minute time limit and he can't phase individual body parts.
Along with Invisibility, this is one of the main abilities of a fully-realized Void hero in Homestuck. Roxy, the only Void hero known to earn these powers, can currently only use them with the help of the Black Queen's ring, but it's implied that any God Tier Void hero can use them at will.
In the Whateley Universe Phase (Ayla Goodkind) has the ability to become a number of variants of intangible, most notably fully intangible, nearly-intangible, or super-dense, so he's Nigh Invulnerable. This is supposed to be an extra-dimensional density-changing ability (though according to the school testing wonks, it's not, but the ensuing scientific babble is hard to understand). While intangible (or close to it), he can fly, and at his most dense, he is almost impervious to damage. But when he first manifested as a mutant and got his powers, he couldn't control them and had trouble NOT sinking through the floor, or going super-dense and smashing his bathroom to shards. Or, for that matter, not going intangible and leaving parts of his clothing behind.
He also has a fun/hilarious Required Secondary Power: if he becomes solid while phased through something, the other object disappears. Forever. This results in lasting Clothing Damage (and, y'know, other damage) if he phases in and out quickly, but turns a scary situation into lots of awesome when he has an I-beam stuck through his chest as part of his Test To Destruction experimentation by his family's anti-mutant science labs.
Fine Structure has Mitchell Calrus, "The Four-Dimensional Man", who is able to turn intangible at will.
Mitch is notably not able to breathe while phased and makes use of scuba equipment.
The Wraith is a mercenary thief from the Global Guardians PBEM Universe. He specializes in getting into places that others cannot get into by way of this power.
Phantom Panther is a Chinese martial artist who can not only turn intangible but can become invisible as well.
Several of the speedsters in the setting can vibrate through solid material.
A-GENT 66 can become Intangible at will and pass through various materials. Except lead, for a yet unexplained reason.
In Worm, Shadow Stalker's shadow form can pass through ordinary obstacles.
A dark power in Phaeton, intangibility allows for full transvection when partially inside something, preventing the user from passing through the ground but means that if they don't see something coming it will not pass through them, alternatively it is achieved through tiny short range wormhole generation.
Buzz Lightyear Of Star Command: Voluntary intangibility is an inherent ability of the Tangean people, including ranger Mira Nova. It's more often used to walk through walls and even phase through floors (in-universe it's called "ghosting"), but on a handful of occasions, it's been shown to allow projectiles to pass right through Mira without harming her. On one occasion, she ghosts into Buzz's body. Ew.
Mind-reading is another, lesser-used, Tangean ability, in which Mira ghosts her hand into the subject's head.
Batman Beyond: An unscrupulous reporter stole a device that gave him this power so he could spy on anyone he pleased and make a fortune as a celebrity gossip columnist. In the end, he subverted the trend and did suffer the Karmic Death of becoming permanently intangible and falling first through each individual floor of the building he's in, then falling through the basement into the ground, and then, presumably, the Earth's crust all the way to the core. Depending on whether or not heat/gravity has any effect on him, and how long he can survive in an intangible state, it may be a horrific And I Must Scream ending of the kind Beyond loved to spring on the audience.
Since the intangibility is simply an effect granted by a full-body suit, it's likely that at most he has only three days before he starves/dehydrates to death in the suit, assuming it's internal air supply (which you would expect it to have, solving the "how do you breath" issue) lasts that long.
The lead character of Danny Phantom has this as part of his ghost powers, as well as flight (solving some problems) and invisibility (introducing new ones).
Ghostfreak in Ben 10. At one point, someone manages to turn the intangibility off, leaving Ghostfreak stuck in solid form temporarily.
Xiaolin Showdown: SERPENT'S TAIL!! It even becomes a plot point when Wuya combines its effects with the Reversing Mirror to become tangible again. Why this actually brought her back to life instead of just making her a solid ghost is left entirely unexplained, though.
Neutrinos act very much like this, as they can (and do) pass through people, buildings, and even entire planets without interacting with anything in their path. It takes a very sensitive detector in large tanks of very pure water deep underground to detect their presence, and even then the massive burst of neutrinos from a supernova yielded only about 6 confirmed interactions.