In this chapter, we are going to study the basic material of martingale theory for discrete parameter. Although martingales were already discussed by P. Lévy in 1937, without explicitly naming them, it was J. L. Doob in the 1940s, who not only named them but realized their potential and did the fundamental development of the theory. As we discussed in Chap.  1 , several mathematicians worked on objects that turned out to be martingales before the theory was developed; for example, R. E. A. C. Paley [134] and J. Marcinkiewicz proved inequality ( 1.26 ) for the HaarHaar functions and Walsh systemsWalsh functions which are special cases of the results of D. Burkholder in his famous 1966 paper [29]. In a different direction, in 1928 R. Courant, K. Friedrichs and H. Levy [60] used ideas related to the notions of martingales, although without randomness, to study harmonic functions, in the papers which introduced the finite element method for the numerical approximation of solutions of partial differential equations, obtaining the now famous Courant–Friedrichs–Lewy.