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In the Mario franchise, a wish is a strong hope or desire. Wish-granting is tied to Stars, with Super Mario RPG: Legend of the Seven Stars even revealing that wishes turn into physical Wish Stars when they reach the Star Road, which then fall back to the earth as shooting stars after they are granted. In Paper Mario, it is revealed that the Star Spirits choose which wishes to grant based on the integrity of the wisher, with selfish wishes never being granted. Wishes have also been shown to have a direct effect on the world during some of Mario's adventures, such as counteracting and weakening the powers of the Shadow Queen in Paper Mario: The Thousand-Year Door. It was also speculated by the Star being Geno that without the power of wishes and the Star Road, Mario might be unable to rescue Princess Peach from Bowser.
Super Mario RPG: Legend of the Seven Stars
During the events of Super Mario RPG: Legend of the Seven Stars, the Star Road is shattered into seven Star Pieces by the giant sword Exor as part of the invasion by the Smithy Gang from another dimension. As a result, no wishes can be granted: no new wishes are made into Wish Stars, and existing Wish Stars fall down to Star Hill despite not being granted. The situation was so dire, the stars sent one of their own, Geno, to assist Mario in collecting the pieces of the Star Road. As explained by Geno, without the Star Road granting wishes, peace could not return to the Mushroom Kingdom so long a theirs was "a world where wishes can't come true anymore."
Along with Peach, Bowser and Mallow, Mario and Geno collect six of the Star Pieces and confront Smithy, who has the last piece. Upon learning of the power of the Star Road, Smithy resolves to keep the piece for himself, and replace wishes with weapons as the dominant force in the world. In the end, however, Smithy was destroyed in battle and the Star Road was repaired, and the power of wishes was restored to the world.
Paper Mario series
Paper Mario reveals that the power of wishes has existed in some form since the beginning of time, with the Star Spirits in Star Haven watching over the world below and granting wishes to deserving people, thus allowing the world to prosper peacefully. Because Bowser's wishes were always selfish, they were never granted, and in the outset of the game, he attacks Star Haven and steals the powerful Star Rod in order to grant them himself. Using the power of the Star Rod, he traps the Star Spirits in playing cards, kidnaps Peach and her entire castle, renders himself invincible and even defeats Mario initially. Fortunately, Mario is able to free the Star Spirits and face Bowser once more, and once Peach wishes for the Star Spirits' power to increase, Mario is able to defeat Bowser using the Peach Beam, and the authority to grant wishes via the Star Rod was returned to the Star Spirits.
Paper Mario: The Thousand-Year Door
Wishes play a much smaller yet still vital role in Paper Mario: The Thousand-Year Door. During the final battle, the Crystal Stars fly across the land and begin transmitting the wishes and positive energies of the people he met on his journey, which then weaken the Shadow Queen enough for Mario and his partners to be able to deal damage. Before the wish powers intervene, like Bowser in the last game, the Shadow Queen is invulnerable to attacks.
Paper Mario: Sticker Star
The Sticker Comet holds the wishes of all the citizens of the Mushroom Kingdom. At the start of the game, Bowser attempts to steal the comet to grant his own wishes, causing it to shatter into Royal Stickers. At the end of the game, once the comet is restored, Mario wishes for Kersti to be brought back to life, while Princess Peach comments on how disastrous the power to grant wishes without wisdom can be.