This article presents an optimal shape design methodology for heat conduction and convection problems. In this study, the shape of the conductive and convective medium is parameterized by means of non-uniform rational B-spline (NURBS) surfaces, and their control points represent the design variables. The conductive and convective domain is discretized by choosing the parameters of NURBS surfaces as generalized curvilinear coordinates, and the heat conduction and convection equation is solved using the finite difference method. The simplified conjugate-gradient method (SCGM) is used as the optimization method to obtain the optimal shape and adjust the design variables intelligently. By optimizing the profile of a straight fin with the objective of enhancing heat transfer rate and reducing the fin mass the methodology is demonstrated for conduction problems and by optimizing the shape profile of a natural convective cavity with the objective of reducing the maximum wall temperature the methodology is shown for convection problems. Manuscript profile