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Punching is an attack that most characters can do in the Mario franchise.
Super Mario RPG: Legend of the Seven Stars
The punch move is first seen in Super Mario RPG: Legend of the Seven Stars, as Mario's move to attack enemies and bosses when he is unarmed or when he is equipped with either the Punch Glove or Mega Glove. If the player performs a Timed Hit, Mario will do an uppercut after the first punch. Other party members (such as Mallow when unarmed, or equipped with the Whomp Glove or Sticky Glove) and enemies (such as Terrapins) can punch as well, and Terra Cottas and the boss Jagger have a special attack called Terrapunch which involves them dashing into one of the party members with a punch to damage them.
Super Mario 64 / Super Mario 64 DS
Arguably the most famous appearance of the punch is in Super Mario 64, which is Mario's basic attack. Pressing the same button will make Mario throw another punch, and pressing it a third time will make him kick instead, though he kicks by default when he is in the air. These punches are used to attack enemies, destroy blocks, and grab crates. Luigi and Wario can also use punches in the remake, Super Mario 64 DS, as well as Yoshi once he acquires one of the other characters' caps. While each character's punches are similar to each other in execution, Wario's punches can be used to destroy Black Bricks.
Donkey Kong 64
One of Chunky Kong's special moves in Donkey Kong 64 is called the Primate Punch (scientifically named Sandwichium Knucklus), which can be purchased from Cranky's Lab for five Banana Coins. This move is activated by pressing and holding the button while hitting the button when Chunky Kong is immobile, which makes him throw a powerful punch that can take down most enemies, destroy boxes and gates, and activate wall switches.
Super Smash Bros. series
Most characters have punching moves in the Super Smash Bros. series. In most cases, these moves are standard, tilt, or smash attacks, though some, such as Captain Falcon's Falcon Punch or Donkey Kong's Giant Punch, are special moves.
Mario Party series
Various Mario Party minigames involve punching objects or other players. Some characters have alternate animations for their punch, most noticeably Peach and Daisy, who have an open-hand slap instead. Yoshi uses a headbutt in the Nintendo 64 Mario Party games, but it was changed to a punch in all later installments. Birdo also has a slapping animation similar to Peach and Daisy in Mario Party 7, but it was also changed to a punch in Mario Party 8. In Super Mario Party Goomba headbutts due to having no arms.
In Wario World, Wario punches when the player presses the Button. By holding the Button down, a punch followed by a Dash Attack will occur. While beating up an enemy or a group of enemies, every third punch will be a headbutt, which is three times as strong as a punch.
Mario & Luigi series
In Mario & Luigi: Bowser's Inside Story and its 3DS remake, Mario & Luigi: Bowser's Inside Story + Bowser Jr.'s Journey, one of Bowser's moves in the overworld is punching, which can be used to break or move objects, and activate switches. In battle, he winds up a punch that can be executed with , and depending on how far back Bowser winds up, the move deals extra damage. Later on in the game, he gains the Sliding Haymaker, which allows him to move while punching, allowing him to break much more durable objects and move across large gaps without falling, similar to the Spin Jump that Mario and Luigi use). Bowser (as well as Bowser X, a much more powerful version of him fought in the Gauntlet) also punches as one of his attacks as a boss later in Toadley Clinic, as well as in Dream's Deep in Mario & Luigi: Dream Team.
Wario Land: Shake It!
In Wario Land: Shake It!, Wario can perform a punching move called the Earthshake Punch when the Shake Meter is full. Once used, Wario will punch the ground, releasing a shockwave that stuns all enemies on the ground, cause unsteady blocks to fall, or activate pillars, though the meter is fully depleted after one use. The Shake King uses a similar move during his battle with Wario, which involves him using both of his fists.