||This paper is a survey on American option pricing theory. The first chapter is an introduction to American option. The second chapter is on Black-Scholes equation. The third chapter is numerical procedure for American option, binomial tree is introduced and a computer program gives the value of an example option following binomial tree. The forth chapter is focused on American option on asset paying discrete dividends, for American call paying arbitray times of discrete dividends, analytical pricing formulae are given. Computer programs show option values both by the binomial tree and analytic formula.