Abstarct: In this thesis‎, ‎first we have introduced classical random walk on a line and we have calculated probability distribution and standard deviation of it’s distribution as well‎. ‎‏‎T‎hen the quantum walk on a line introduced and it's variance and probability distribution compared with classical random walk‎. ‎In the next chapter‎, ‎we have studied two-dimensional quantum walk and we have introduced three famous two-dimensional quantum coins‎. ‎T‎he probability distribution‎, ‎standard deviation and variance of these famous coins have been compared by numerical simulation‎. ‎Furthermore‎, ‎we investigate properties of two kinds of two-dimensional quantum walk (nonreversal and nonrepeating quantum walk) where have been introduced recently‎. ‎Beside‎ ‎the study of variance and the probability distribution for these two new kinds of two-dimensional quantum walk‎, ‎we analytically show that the joint moments is independent from the initial states for nonrepeating quantum walk‎. ‎At the end‎, ‎we have presented our model of two-dimensional quantum walk (clockwise and counterclockwise quantum walk) in such away that the walker does not turn to the left or right in clockwise and counterclockwise quantum walk respectively‎. ‎The probability distribution and variance as well as joint moment’s for these new quantum walks have been studied and compared with other two-dimensional quantum walks‎.