From the Super Mario Wiki
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In Super Mario Galaxy, Grand Stars are used to provide power to the six domes onboard the Comet Observatory, thereby allowing the player to reach more and more galaxies as the events of the game progress. In Super Mario Galaxy 2, Grand Stars still allow the player to reach more and more galaxies, but they now provide direct access to the next World via their transformation into a Portal instead.
Super Mario series
Super Mario Galaxy
Super Mario Galaxy is the game in which Grand Stars make their first appearance. Bowser has taken all of the Grand Stars and is using them for his own evil intentions. As Mario recovers them, he gradually gains access to different parts of Rosalina's Comet Observatory, as the Grand Stars are the power source of the Comet Observatory's Beacon, which in turn sustains all life onboard the observatory. As each Grand Star is found, a path to the next dome is created. It is possible to only have 5 out of the 7 Grand Stars collected to reach the final battle, since Bowser's Galaxy Reactor, where the final battle is, can be unlocked after completing the fourth dome if the player has 60 or more Power Stars.
Super Mario Galaxy 2
Grand Stars make another appearance in Super Mario Galaxy 2, the 2010 sequel to Super Mario Galaxy. In this game, the Grand Stars are once again under the control of Bowser and Bowser Jr. This time around, Grand Stars restore power to Starship Mario, and also create the Portals leading from one World to the next. In addition to this, the Grand Stars in this game are apparently responsible for Bowser's huge size, as he has eaten them to make himself considerably larger. After Mario defeats one of Bowser Jr.'s machines or henchmen, it will explode and release a Grand Star. After Mario defeats Bowser, he'll cough up a Grand Star. During the final battle in Bowser's Galaxy Generator, he is actually able to swallow the Grand Star again, making himself a great deal larger than he had been, and leading to one final segment of the battle. The Grand Stars seem to have an interesting side effect when misused, one that causes Bowser to become extremely tiny after the final battle, as seen in the end credits of the game.
Super Mario 3D World
Mario Kart series
In Mario Kart Wii, Mario Kart 7, and Mario Kart 8, a Grand Star appears as Rosalina's emblem. In Mario Kart 8, a floating Grand Star replaces the star icon and one can be seen rotating on top of the tower in the city in N64 Rainbow Road, and Baby Rosalina's emblem features a Grand Star inside blue bonnett frills.
Names in other languages