From the Super Mario Wiki
“We got a lot of people in Brooklyn. We got pool players, we got beautiful girls, we got restaurants, we got carpenters, plumbers, but normal? Huh-huh! Forget about it!”
One of its first mentions in an actual game was in Mario's Time Machine, in which upon being asked if he was from Italy (due to his accent) by an imperial ruler in China, Mario says that he is from Brooklyn, New York. Of course, granted the time period, the feudal lord didn't know what Brooklyn was.
The Super Mario Bros. Super Show!
In The Super Mario Bros. Super Show!, it is shown that Mario and Luigi ended up in the Mushroom Kingdom by getting sucked down a Warp Pipe connected to the drain of bathtub that they were unclogging on a plumbing job.
In the cartoon episodes, Mario and Luigi often make references about their former lives Brooklyn, or lament on how they wish to return home. The live action segments all took place at the Mario Brothers Plumbing shop in Brooklyn.
In the episode "Brooklyn Bound", Mario and Luigi meet up with Salvador Drainado, a fellow Brooklyn plumber who had been lost in Mushroom Kingdom for thirteen years. However, Salvador Drainado had discovered a one-way drainpipe to back to Brooklyn, which could only be used once. With Mario and Luigi's help, Salvador Drainado was able to reach the pipe, and offered Mario and Luigi to come along. Mario and Luigi almost go along with Salvador Drainado, but decide to stay behind in the end to save Princess Toadstool and Toad from Koopa Khan and the Koopa Pack.
Brooklyn is also featured in the episode "Plumbers Academy". In an extended flashback sequence, Mario tells the story of how he and Luigi became plumbers after being flunked by Sergeant Kooperman at the Brooklyn Plumbers Academy.
In the episode "Flatbush Koopa", a Warp Pipe leading to Brooklyn is discovered, and Mario and Luigi, having defeated King Koopa, get to return home. When they get there, they are surprised to see that King Koopa is slowly turning the city into "Kooplyn", as revenge against the Mario Bros. for ruining all of his evil plots. Although they manage to foil King Koopa and lead him back to the Mushroom Kingdom, Mario blows up the pipe with a Bob-omb, if only so that King Koopa can't return to Brooklyn. By blowing up the pipe, Mario and Luigi are forced to stay in the Mushroom Kingdom.
The Adventures of Super Mario Bros. 3
In The Adventures of Super Mario Bros. 3, more Warp Pipes leading to Earth had been discovered, making access to Brooklyn much more convenient.
In the episode "Toddler Terrors of Time Travel", Bowser and Kooky try to go back in time to stop the Mario Bros. from going down the drainpipe in Brooklyn, but their plan screws up when the Time Tube turns them, along with Mario, Luigi and Toad into babies.
In "Misadventures in Babysitting", Mario and Luigi accidentally warp into a Brooklyn home, and get suckered into babysitting a brat named Junior, who warps himself to the Mushroom Kingdom in a game of hide and seek.
In "Recycled Koopa", the Koopalings dump their garbage down a Warp Pipe to Brooklyn. As a result, the garbage mutates the people there into Koopa Zombies. Mario and Luigi do what they can to clean up the garbage before the entire city gets contaminated.
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