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The Phantom Thief is a special class of criminal. An elusive and fantastic thief who can be long gone before the victim even knows what hit him. Just as likely to be an antihero as a villain.
There is no single set model of the Phantom Thief, but they usually have some of the following characteristics.
Phantom Renegade of Medabots steals rare medals in order to keep them out of the wrong hands. He would actually do a pretty good job, if he didn't clumsily lose them right after gloating over his successful heist.
Robas from Kiba, who becomes the thief "No Face" because he's bored and wishes for a challenge. It's also hinted he may be trying to help out his family financially, since their fortune is somewhat on the rocks.
Kamen Rider Diend is actually named has Daiki Kaitou, but he's only partway to being a full-fledged Phantom Thief. He has the "theft of valuable objects" and "uses trickery and illusions" bits down but his personality boils down to "real-life Troll" for his victims (especially Tsukasa), and he steals not out of a sense of social justice but because he enjoys it (though that's just his assertion, in truth it's because he's trying to fill the void left behind by his brother being brainwashed by a crime boss).
Guess what: he really isn't. Further guess what: he really is but he didn't commit the theft in question - that was Atmey, who was responsible for putting him up to most of his previous heists and was using this one as an alibi.
It's an awesome moment when you get the loveable little bastard cleared of all the crimes he actually did commit, almost by accident.
Ace Attorney Investigations has another in "The Great Yatagarasu", a sort of Intrepid Reporter version of the Vigilante Man. The Yatagarasu has two odd quirks compared to other phantom thieves. First; the Yatagarasu doesn't steal much that's tangible. Rahter, the thief makes off with evidence of dirty dealings and makes them public. Secondly, the Yatagarasu doesn't actually send calling cards to the victim beforehand. Rather, the victim doesn't know that their info was even stolen, until it shows up on the news, having been delivered to the authorities alongside a calling card. Actually, there's three of "him", all Cowboy Cops. One however is The Mole planted in the group to keep them away from a smuggling ring. They are Byrne Faraday, Detective Tyrell Badd and Calisto Yew (She's the mole.)
In the Thief series, Garrett is a classic Phantom Thief and Anti-Hero, though he steals for his own gain as well as for the thrill of it.
The higher difficulty levels impose a code of honor (no killing humans; zombies and the like are fair game). The Fandom takes it further still, with a common Self-Imposed Challenge being the Ghost run - a playthrough in which Garrett gets in, steals everything not nailed down, and gets out again without anyone even suspecting he was ever there.
To clarify, Ghosting means leaving no evidence anyone was there. No unlocked doors (lock them again), nobody looked round and asked "What's that noise?", no broken windows, no doused torches (in Thief 2, which this is very popular for, re-lighting torches is easy) and definitely nobody injured or killed. The only difference is all the loot is gone.
Skye from Harvest Moon DS Cute is widely known as "Phantom Skye". He doesn't seem to steal for any other reason other than for the fun of it (or, should you choose to woo him, to meet with the player).
Actually, Skye steals because he wants to make really good curry. The items he steals are ingredients and other people's curry dishes (to taste them).
Wario: Master of Disguise features Wario becoming one of these using a magic wand he took from a real phantom thief. The game's original title, Kaitou Wario the Seven, is clearly meant to suggest both Lupin the Third and the Fiend of Twenty Faces (since seven is actually how many costumes Wario has, besides the basic thief).
MapleStory gets one as a playable character, unsubtly named Phantom.
The Elder Scrolls IV Oblivion has the Gray Fox, who people believe can slip through doors and become invisible. And no one knows who he is, because he's wearing a cursed Daedric artifact that strikes the wearer's name from history.
The newly-introduced Trickster class in Fire Emblem: Awakening, a promotion of the standard Thief, is literally described as a glamorous phantom thief. Its members are capable of using healing staves as well as swords.
Rhythm Thief And The Emperors Treasure gives us the aptly named Phantom R, a thief known for stealing priceless works of art... only to return them days later without a scratch. The Police always fail to catch him, even after they spot him in the act. Then again...
Geist from the webcomic Heist. He lives in a superhero universe and only his former clients even know he exists.
The unknown rook (thief) in Snow By Night, who goes after objects having to do with hearts. Even the other rooks are baffled by this person. Turns out the thief is Snow By Night, a manitou with magic powers, which explains why she is so hard to find.