Abstarct: Decoherence is a physical phenomenon that arises from the interaction of quantum systems and their environment. Decoherence used to be thought of as an annoyance as it used to be equated with loss of quantum information. However, it has been found that decoherence can indeed play a beneficial role in natural processes. The quantum walk is the quantum version of the classical random walk. The superposition property of quantum systems allows us to define quantum walker which can walking in all possible path, simultaneously and the interference of these paths makes some important differences between the quantum walk and the classical random walk. Notable differences between the quantum walk and classical random walk are the quadratic dependency of the variance on the number of step and the complex oscillatory probability distribution in the quantum walk instead of the linear variance dependency and the binomial probability in the classical random walk. In this thesis, we introduce and study classical random walk and the quantum walk on a line. We will compare probability distribution and variance of quantum walk with classical one. We introduce the Fourier transformation as a standard method to map the evolution operator into k space. In chapter two, we will introduce Kraus operators method and analytic formula for variance which is baxsed on this method. We will find variance of various types of channels as examples of coin decoherency. In chapter three, by introducing general form of variance formula the effects of noise in coin space, position space and coin-position space has been studied, we will see that the existence of noise (even a weak noise ) in coin space will eliminate quantum properties, while noise on position space as a tunneling to nearest neighbor can preserve it. The preservation of variance in tunneling decoherncy motivates us to check other types of tunneling which we deal with it in chapter four. We have introduced most general form of tunneling effects in which the walker will be able to jump to any other neighbors and show that this type of decoherency not only doesn’t fade quantum properties but also increase variance a bit.