In this thesis, the buckling of simply supported and clamped Functionally Graded super elliptical plates is investigated. The periphery shape of plate is defined by a super elliptical function with a power, corresponding to shapes ranging from an ellipse to a rectangle. Such a plate shape has practical applications, as the advantageous curved corners help to diffuse stress concentrations. The loading considered for the buckling problem is that of in-plane uniform pressure along the periphery. Functionally graded materials (FGMs) are one of the advanced high temperature materials capable of withstanding extreme temperature environments. FGMs are composite and microscopically inhomogeneous in which the mechanical properties vary smoothly and continuously from one surface to the other. This is achieved by gradually varying the volume fraction of the constituent materials. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the equilibrium and stability equation and the total potential energy are obtained for Functionally Graded Plates (FGP). The computations are carried out by both of the Rayleigh-Ritz and Galerkin method. Rayleigh-Ritz use is based on the Trefftz criterion, which defines the critical load as the smallest load for which the second variation of the potential energy assumes a stationary value. The Galerkin?s method is proved to provide solutions of significant accuracy with minimal manual effort and computer execution time for various problems.In the Galerkin?s method, accuracy of the results depends on the selected shape function. Converging solutions can be obtained much faster by satisfying all of the geometrical and kinematical boundary conditions of the problem by the assumed series expression of the plate surface. Galerkin?s method can be applied to small and large deflection theories, linear and nonlinear problems of plates provided that differential equations of the problem under investigation have already been determined. The study is performed for a wide range of super elliptical plates. The critical buckling loads are obtained for a super elliptical plate with different super elliptical powers, various powers of FGM and some aspect ratio. It is concluded that, the equilibrium and stability equations are identical with the corresponding equations for homogeneous isotropic plates, the critical buckling load for the functionally graded plates is generally lower than the corresponding value for homogeneous plates. Functionally graded plates have many advantages as a heat resistant material, but it is important to check its resistance to buckling, the critical buckling load for the functionally graded plates is reduced when the power law index k increases, the critical buckling load for the functionally graded plates decreases with increasing aspect ratio a/b, the critical buckling load for the super elliptical plates decreases with increasing power of supper ellipse n. By equating power of FGM to zero, predicted relations are reduced to the relations of homogeneous plates. The detailed parametric studies are carried out to investigate the influences of aspect ratio, super elliptical power and power of FGM on the critical buckling load. Keywords: Buckling, Super elliptical plate, FGM, Rayleigh-Ritz method, Galerkin method, Classical plate theory