The effects of boundary perturbations on eigenvalues are reviewed. The perturbation theory is developed for application to calculations of the buckling of reactors whose lateral surface is shaped like a right circular cylinder or a sphere. It is shown that with the perturbation approach applied, the zeroth-order approximation can be a circular cylinder or a sphere of such a radius that the first-order correction for the buckling is zero. A buckling formula for reactors with a cylindrical side surface has been obtained within the framework of the second-order perturbation theory. An elliptical cylinder and regular polygonal prisms are reviewed for illustration purposes.