Kame Game is the name of Solomon Muto's shop. The shop sells various games. Solomon, Yugi's mother and Yugi live here together. The word "kame" is Japanese for "turtle". In the dub, the shop is referred to simply as "Yugi's grandfather's game shop" and is never revealed to have a name, only having the word "game" written on the front.
The shop also appears to be the home of the Muto family, having several upstairs bedrooms and other living quarters shown on occasion.
Yugi found the Millennium Puzzle in the basement of Kame Game when he was a child. Eight years later, he solved the puzzle in his bedroom, while under pressure to figure out what to do about Ushio's self-imposed bodyguard fees. (In the first series anime, he didn't insert the last piece until he was at school.) He then became possessed by Yami Yugi for the first time.
Yugi took his friends here to show them his grandfather's "Blue-Eyes White Dragon" Duel Monsters card one day. Seto Kaiba also entered the store that same day and saw the card. Grandpa refused to sell or trade the card to him, but Kaiba later made other unsavory attempts to get the card afterwards.
A number of years later, Jaden Yuki and his friends visited Kame Game on a trip to Domino City. Solomon Muto was still running the shop, despite his advanced age.