Functionally Graded Materials (FGMs) are special class of composite materials that are composed of two different materials. In these materials mechanical and thermal properties vary smoothly through the thickness or other directions. In this thesis elastic-plastic buckling of a FG thick rectangular plate under biaxial loading is studied. The analysis is based on the both common plasticity theories; Incremental Theory (IT) and Deformation Theory (DT). After presenting elastic-plastic buckling relations for a homogenous plate, the formulation is redeveloped for a FG plate. Using the derived governing equations, elastic-plastic buckling analysis is conducted for the FG material case study. A metal-metal type (Aluminum-Steel) was selected as the FG case study. The mechanical properties, such as elastic modules, density and Romberg-Osgood parameters, is assumed to vary as a power function through the thickness of the plate. The Mindlin plate theory or First Order Shear Deformation Theory (FSDT) has been utilized to consider shear deformation. As it is known, in FSD plate theory the shear strain is assumed to be constant through the thickness, and then, a correction factor is applied to modify the non-zero shear strain on the top and bottom surfaces of the plate. The minimum total potential energy functional (uniqueness criterion) is used to determine the critical buckling stress. Based on Rayleigh-Ritz method, the displacement fields are approximated using complete polynomial functions which involve unknown coefficients. The geometric boundary conditions are satisfied using appropriate base functions, multiplied into the approximation polynomials. The dimensionless form of parameters is introduced to rederive dimensionless form of the equations. This enables the analysis to represent more comprehensive results of the buckling study. By substituting the approximate functions of displacement field into the integral uniqueness criterion, the integral is calculated as a function of the unknown coefficients of the displacement polynomials. The integral result should then be minimized with respect to these unknown coefficients to determine the critical buckling stress. This minimization procedure results in a homogeneous system of equations with respect to unknown coefficients. For nontrivial solution (zero value for all unknown coefficients) the determinant of the coefficient matrix of the equations system should be equal to zero. A nonlinear algebraic equation is then obtained in which the boundary traction loading is unknown. The smallest root of this equation is the critical buckling stress. In this thesis, a computer code has been developed in MATHEMATICA software to implement the solution approach and calculate the critical buckling stress. To validate the analysis, the convergence of results with respect to the polynomial order is first investigated. A fifth-order polynomial was found to be adequate to obtain sufficiently accurate results. Then the buckling stress for a homogeneous thick plate has been calculated and compared with previously published results. A very close agreement was observed. After the validation was carried out, the effect of important factors on the buckling of FG thick plate is investigated. These factors involves thickness ratio, aspect ratio of the rectangular plate, boundary conditions, the loading ratio, and the exponent parameter in FG model. The effect of FG exponent parameter was found to be insignificant. This is due to relatively close values of the mechanical parameters for the two materials which constitute the FG plate. On the other hand, the loading ratio was found to be the most important factor which may affect the critical buckling stress. Keywords: Elastic-Plastic Bukling, FG Thick Plate, Incremental theory, Deformation theory, FSDT