For the solution of elastic-plastic buckling problems, the variational methods (Rayleigh, and Galerkin methods) have been the most powerful and useful to engineering purposes. In the recent years, the developement in high speed digital computers has been giving favor to a numerical approach that is referred to as “the finite element method”. Gallagher first applied this method to buckling of columns and Kapur established the basic approach in obtaining the solution of elastic plate buckling with the finite element method. Based on the results of the investigations mentioned above, the authors have developed the finite element method and succeded in applying the method to the solution of elastic-plastic buckling of plates with the aid of the plastic theories, the plastic deformation and the plastic flow theories. In the paper, the accuracy of the solution by the method was examined for both elastic and plastic buckling of plates at a relation to the number of finite elements divided in the plate. As a basic example of the application, elastic, elastic-plastic and plastic buckling of plates containing residual stresses was solved by the method and compared with the analytical solution obtained by the energy method. The result was quite satisfactory and suggested the usefulness of the method.