Numerical methods for engineering and scientific problems have improved considerably due to quick advances in computer technology. The meshless methods are modern numerical methods, which are implemented only based on a grid of arbitrarily distributed nodes, without any finite element mesh. One of the most recent meshless methods is the Finite Point Method (FPM). The major difference between this method and other meshlessmethods is satisfaction of differential equations and boundary conditions in FPM, which is based on point collection procedures. This type of implementation has considerably improved the efficiency and flexibility of the finite point method. Instability of the FPM is the major disadvantage of the method , which has restricted its application and involved many researchers in development of stabilized finite point methods. Based on above, one of the major goals of this research work is to develop a Finite Point Method with improved convergence and stability in compariso with standard version of the method. In this work we have presented a modified version of the Weighted Least Square(WLS) approximation, by which the approximate function passes exactly through central point of each cloud. The concept of mapped clouds is proposed in order to improve the quality of approximation in some non-homogenous grids. A brief discussion on effect of cloud shape on approximation quality for different grids is also presented. In addition to above, the Weited Residual Finite Point Method (WRFPM) is one of the major results of the present research. The method is based on satisfactio of both govering differential equations and Neumann boundary conditions simultaneously using the weighted residual formulation. The improvementin convergence and stability of the method is observed in results obtained from numerical examples. A new systematic algorithm is presented, by which the FPM might be implemented systematically. The algorithm also benefits from the symmetry of final discrete equations.