The decomposition method first proposed by Adomian is an effective procedure for a semi-analytical solution of a wide range of dynamical systems. It is based on decompositions of the operator and the solution, and does not require linearization, or weak nonlinearity assumptions, closure approximations or perturbation theory. A well-known, long-standing problem in reactor kinetics is the stiffness arising from the orders of magnitude difference between the prompt and delayed neutron lifetimes, which results in the restriction of very small time step increments in numerical solutions to the kinetics equations. There have been a number of methods, for example, stiffness confinement method (SCM), singular perturbation method, and $\theta$ weighting method, suggested to avoid the difficulty, but they generally involve some approximations or apply effectively only to certain types of problems. In this thesis, the Adomian``s decomposition method (ADM) was applied to several reactor kinetics problems: step reactivity insertion, ramp input of reactivity, and reactivity feedback nonlinear problems. The results obtained with ADM are far better than those of other methods. Since in ADM the solution is decomposed into the Adomian polynomials, we can use large time step increments. Once the model of the dynamics system is given, the Adomian polynomials can be generated recursively. And in the case of input data change but for model change, we do not need to generate the Adomian polynomials again. Thus, ADM is very efficient and accurate. Since ADM does not require linearization or perturbation, it is particularly useful for nonlinear problems. Also if one has some idea of an analytic solution that approximates the solution of the problem to be solved, he can obtain the solution very fast by introducing some transformation of variables and then applying ADM.