Generally we have two types of phase transition: Thermal phase transition and Quantum phase transition. Thermal phase transitions occur due to the competition between energy and entropy of a system at finite temperature. In contrast, quantum phase transition occurs at absolute zero temperature and they are driven by quantum fluctuations.. Study of the quantum phase transitions in the absolute zero temperature is a necessary step in the complete understanding of properties of materials at higher temperatures. In this research we study the quantum phase transitions in the spin systems. One promising methods for studying spin systems is series expansion method. Series expansion method consists of high temperature and low temperature series expansions and cluster expansion. In this thesis we study the cluster expansion method. Cluster expansion method is a perturbative method based on non-degenerate Rayleigh-Schrodinger perturbation theory which is developed for performing high order perturbation series.. Using cluster expansion method, it is possible to perform high order expansions for quantities like ground state energy, magnetization, susceptibility and other quantities. In cluster expansion method the problem of performing expansions up to certain order for infinite system is reduced to performing series for finite number of small clusters. After we obtain series, we extrapolate those using standard series analysis methods.. At the end we study the application of the cluster expansion method in two spin systems. First we study the antiferromagnetic Heisenberg model with nearest and next nearest neighbor interactions on honeycomb lattice. We obtained series for ground state energy, magnetization and parallel susceptibility up to eighth order, and then we extrapolate series for magnetization and obtain a quantum phase transition where magnetization vanishes. Second we study the one dimensional quantum Ising model with frustrated nearest and next nearest neighbor interactions in a transverse magnetic field. This model has three well known phases, which are ferromagnetic, paramagnetic and anti phase, and one relatively unknown phase called the floating phase. Ferromagnetic phase and anti phase are the ordered phases of this model for weak transverse field and paramagnetic phase is disordered phase of this model for a strong transverse field.. We obtain perturbative expansions for ground state energy, magnetization and parallel susceptibility around the ferromagnetic . We expect the critical exponents of ferromagnetic-paramagnetic quantum phase transition in one dimensional quantum Ising system be the same exponents related to the dir=rtl Keywords: Quantum phase transition, Quantum fluctuations, Series expansion method, Cluster expansion method, Heisenberg model, Transverse Ising model