Abstarct: The issue of searching was always one of the most important problems in different areas of science. searching an unstructured set of N items requires O(N) queries in the worst case. In the quantum domain, however, Grover’s algorithm can perform unstructured search by making only O(√N ) queries, a quadratic speedup over the classical case .Quantum walk is quantum counterpart of classical Random walk which has a different behavior due to the quantum phenomena such as quantum interference, entanglement and superposition. One of the most important applica tions of quantum walks is in quantum algorithms designs. we organize our thesis as follows: first of all, we make a short review of classical and quantum walks. We compare the probability distribution and standard deviation of CRW and QW and show that standard deviation in QW is quadratic while classical one is linear which is the most important difference of these two types of walk.. Following the first chapter we express other differences between these two types of walks and introduce an analytical approach to find quantum probability distribution. Also we mentioned one dimensional quantum walk with generalized coin and quantum walk of higher dimensions. In second chapter one can study a brief review on sev eral quantum algorithms, Grover search algorithm in details and proving that how the algorithm leads in quadratic speedup. In chapter three we deal with search algorithm on a hypercube lattice using coined walk and introduce an approach for optimizing the algorithm. In continue we introduce a new walk with discrete evolution, namely scattering walk in which the quantum particle (walker) takes place on the edges of a graph and the graph vertices act as scattering centers. The evolution operator and state of system after certain number of steps for this type of walk and a unitary equivalence between discrete models has been studied. We show that by using an appropriate initial state and certain number of steps, this type of walk can find a structural anomaly in a graph more efficiently than a classical search can. Then we provide a general approach in star graphs which have high symmetries and show that it is possible to find the target item within a better time interval than classical mode through reducing Hilbert space.