Abstarct: In this thesis, first the concept of classical random walk on the line has been introduced and then quantum walk has been studied. Hilbert space of quantum walk which contains coin space and position space defined. Coin operator instead of coin has been used for quantum walk and its effects on quantum walk have been studied. In the following, we showed that by knowing the initial state and using evolution operator, how the next state can be constructed. It can be shown that by using Fourier transform, position space of random walk can be writing into a two by two matrix. By applying inverse Fourier transform, distribution of quantum walk amplitudes in basis of position has been calculated. In the following asymptotic behavior of quantum walk will been investigated. Moreover, by using the stationary phase theorem distribution of quantum walk amplitudes in basis of position have been investigated. Then another type of quantum walks known as coinless quantum walk has been introduced. This kind of quantum walks first proposed by Patel. Its Hilbert space composed of only position space and the coin space is no longer exist. Two states corresponding to the coin is dumped in to the index of odd and even position indexes. By introducing two evolution operators a quantum walk can be described. By using Fourier transform eigenvalues and eigenvectors and then amplitude distribution of this kind of quantum walk in k-space has been estimated. Again by using inverse Fourier transform, amplitude distribution of this kind of quantum walk in position space can be determined. Asymptotic behavior of this kind of quantum walk have been investigated. All, integral estimated by stationary phase approximation and. Finally, coinless quantum walk compared to coin-baxsed quantum walk and its advantage, weakness, similarities and differences have been investigated.