Luigi Cap

The Luigi Cap (or Luigi's Cap) is one of Luigi's important pieces of clothing. Like his brother, Luigi has worn the hat since he was a baby.

Mario Bros.
In Mario Bros., the Luigi Cap color was taken from the Shellcreeper's palette to distinguish Mario from Luigi. The cap was made green with a green "L" emblem.

Super Mario Bros. / Super Mario Bros.: The Lost Levels
The Luigi Cap was recolored white in Super Mario Bros. and Super Mario Bros.: The Lost Levels.

Super Mario Bros. 2
In Super Mario Bros. 2, the tip of the cap had been changed to black instead of its usual green. This happened for the Mario Cap as well. However, after this game, the Luigi Cap reverted to having a green tip.

Super Mario 64 DS
In Super Mario 64 DS, Luigi's Cap was used to shape-shift into Luigi if the player was either Mario, Wario, or Yoshi. In the same game, Luigi is affected in the same way as Mario and Wario if his cap is lost. Luigi's defense and attack sharply decline while his cap is absent from his head. If the player gets the cap back, Luigi's attack and defense revert to normal. Until Luigi is unlocked, it only appears in Big Boo's Haunt, on any star except the first one. If the player obtains a Power Flower while wearing the cap, they will turn into Vanish Luigi. If he loses his cap and gains a star from another level, Luigi can simply talk to Toad (who will have found and be wearing Luigi's Cap).

Super Mario 3D Land / Super Mario 3D World / Super Mario 3D World + Bowser's Fury
In Super Mario 3D Land and Super Mario 3D World, Luigi, like his brother, loses his cap in his Small form. However, in the former game, once the player has 1110 lives, his regular form loses his cap instead.

Super Mario Odyssey
The Luigi Cap can be obtained for Mario to wear in Super Mario Odyssey. It can either be purchased from a Crazy Cap store or obtained through scanning a Luigi amiibo (the latter also unlocks Luigi's costume).

Mario Kart series
In Mario Kart 64 and Mario Kart Wii, in the Luigi Circuit (Luigi Raceway in Mario Kart 64) courses, giant versions of the Luigi Cap can be seen on top of the grandstands.

Mario & Luigi series
In Mario & Luigi: Superstar Saga, Mario & Luigi: Partners in Time, and Mario & Luigi: Bowser's Inside Story, the "L" on the Luigi Cap is black. The Mario Cap was changed in the same way. From Mario & Luigi: Dream Team onward, they are the usual green and red.

Super Smash Bros. series
Luigi's Cap can be equipped on a Mii Fighter in Super Smash Bros. for Nintendo 3DS, Super Smash Bros. for Wii U, and Super Smash Bros. Ultimate.

Super Paper Mario
In Super Paper Mario, when Luigi is brainwashed into Mr. L, his cap is slightly changed, with the white on his cap being turned to black and the "L" in the middle being turned backwards.

Mario Sports Superstars
Luigi's Hat can be unlocked as a head accessory for a horse in Mario Sports Superstars.

Mario + Rabbids Kingdom Battle
Rabbid Luigi from Mario + Rabbids Kingdom Battle wears a Luigi Cap.

Trivia

 * The Green Hat, a copy of the Luigi Cap, is an item in  that the player's dog can wear.
 * In Animal Crossing: City Folk, and later New Leaf, there is a headgear item resembling the Luigi Cap which the player's avatar can wear. The item is called "li'l bro's hat", in reference to Luigi being the younger brother of Mario.
 * On the North American and European box art for Mario Kart: Double Dash!!, the "L" on the Luigi Cap is backwards. This was likely a mistake by the graphic designer.