Abstarct: In the first chapter we introduce Classical Random Walk (CRW). Probability distribution, standard deviation and diffusion factor are some parameters of CRW will be studied in the same chapter. Then the chapter will continue with Quantum Walk on Line (QWL) and introduce evolution operator for QWL and studying its effect on the walker’s state. It is useful to compare probability distribution and variance of QWL with the classical one. Fourier transform would be an appropri ate tool to map the evolution operator into k space to calculate eigenvalues and eigenvectors of the evolution opertor of quantum walk on line. Chapter 2 introduces another model of quantum walk known as Quantum Walk on a Cycle. We introduce necessary modifications to 1DQW operators to be used in Quantum Walk on a Cycle. The chapter will describe how to solve eigenvalue problems for the operator. Limiting Distribution is an important concept which is introduced and investigated in different situations in this chapter. Mixing Time is another concept to be defined in the chapter and we are going to find an upper bound for it. In the last chapter, we will present our model of quantum walk on a cycle, namely Mobius Quantum Walk. Indeed this is a modification to the previous model, i.e. quantum walk on a cycle. In our model, quantum walk on a cycle has been modified by introducing some kind of rotation around the track on the cycle, char acterized by the parameter α. The chapter shows that α and the extended space for the rotation improve the parameters including limiting distribution and mixing time. We also study the changes made by α in eigenvalues and eigenvectors and the way it affects degeneracy, so we find out how it affects different parameters such as limiting distribution or mixing time. Finally, it is possible to assume that the space for rotation and the spin pace are entangled, which is shown to improve limiting distribution.