A martingale is also known as a fair game, for the following reason. In a gambling situation, if \(X_s\) is the amount of money the gambler has at time s, then the gambler’s expected wealth at time \(t>s\) is also \(X_s\) . This concept of martingale applies equally to discrete and continuous parameter. MartingaleMartingales is an important concept used for studying stochastic processes. In this chapter, after giving the definition and some examples of martingale, the Doob’s martingale process, the optional stopping theorem, Jensen’s inequalityJensen’s inequality, maximum inequality theorem, and martingale convergence theorem are presented. The martingale theory of stochastic integrals and stochastic differential equations, are discussed in later chapters.