Abstarct: In the first chapter of this thesis we first introduce Classical Random Walk (CRW). The probability distribution and standard deviation which are important parameters of CRW are calculated by simulation in this chapter too. Then the quantum walk on a line (QWL) as well as its quantum operator are introduced and compared its probability distribution with classical one. Then we introduced the Fourier transformation which is an appropriate tool to map the evolution operator in K space in order to simplify eigenvalue problem. In addition a method is introduced to make a transition from quantum walk to classical random walk. We also show that we can make a transition from quantum walk to classical random walk by using more coins and reducing the effects of interference among different paths. We have compared the probability distribution of single coin by multiple coins with numerical simulation. In the next chapter we introduce some new kinds of quantum walk which are called quantum walk with memory. In these types of quantum walks, current state of the walker is influenced by its previous state. We explain this type of quantum walk and its differences with ordinary walks and try to study the effect of different types of memory on quantum walks as well. In the last chapter, we introduce a new kind of 1-D quantum walk which is called ”accelerated quantum walk”. In this type of quantum walk, we add another Hilbert space to the system in order to define velocity of the walker which is changeable according to the state of the coin. We modify the conditional shifting operator in order to change the position of the walker according to the velocity of the walker as well, and we study the features of this type of quantum walk and compare it with ordinary one. Existence of the velocity Hilbert space in quantum walk lets us to define and test different types of accelerated QW. In this thesis we introduce and study two kinds of these walks namely ”limited Velocity Quantum Walk” and ”Cyclic Velocity Quantum Walk”.