In this study a new method is proposed to obtain shape functions in the finite point method. The method uses B-splines and NURBS as the base functions instead of the conventional polynomial based functions. For this purpose, the routines for construction of B-splines and NURBS as well as evaluation of their derivatieves are described in details. The reproducing property of the functions and the effects of their use in a finite point method are also examined for one-dimentional problems. The method is further extended to two-dimentional problems. Two forms of approaches are employed in this study. The first one is based on using NURBS for construction of the shape functions defined over the whole domain. The second approach is based on construction of the shape functions locally over sub-domains. Convergence of the solution is studied through several heat and elasticity problems. It is concluded that the convergence of the method is sensitive to the relataive positions of the nodes and monotonic convergence is lost when these two sets of points are close together. Simple remedies are given for elimination of such deficiency through suggestion of some patterns for the grid of knots. It is shown that the proposed method is able to give very accurate results with monotonic convergence.