Chapter 3 Games with perfect information This chapter discusses the games with perfect information (PI). The most seriously played PI games are Chess and Go. But there are numerous other interesting PI games: Checkers Chinese Checkers Halma Nim Hex their misre variants etc. Perfect information means that at each time only one of the players moves that the game depends only on their choices they remember the past and in principle they know all possible futures of the game. For example War is not a PI game since the generals move simultaneously and Bridge and Backgammon are not PI games because chance plays a role in them. However some cases of Pursuit and Evasion can be studied by means of PI games in spite of the simultaneity and continuity of the movements of the players. The chapter focuses on infinite PI games. The theory of infinite PI games is motivated by its beauty and manifold connections with other parts of mathematics. They give also a natural mathematical theory for some garnes of pursuit and evasion. Finite and infinite PI games are also used in model theory and in recursion theory.