Abstract: This paper focuses on the how Game Playing is an important domain of Artificial intelligence .AI is also used in game playing. Various researchers have extensively contributed for computer game –playing methods. Games don’t require much knowledge: the only knowledge we need to provide is the rules, legal moves and the condition of winning or losing Game . Both players try to win the game . so both of them try to make the best possible move at each turn. Shannon’s paper on chess-playing and Samuel’s checkers are considered as land marks in the area of computer game playing. We also explain in this paper major components of game playing program , static evaluation function generator, game playing strategies: i) minimax strategy ii) minimax strategy with alpha – beta cutoffs.