We have studied the scattering of plane waves from cylinders and spheres with complex impedance boundary conditions using the Kirchhoff method and the Luneburg–Kline method. A stationary phase approximation is made in the Kirchhoff derivation to eliminate spurious contributions from the boundary of the insonified region. Terms of order up to (ka)−2are retained. Dependence of the cross section on the complex impedance, the incident wave frequency, as well as on the azimuthal angle is illustrated. The curves are found to obey several near symmetries. Further, the Luneburg–Kline method is found to lead to predictions which are often qualitatively different from those of the Kirchhoff method, and the former is observed to be more reliable. Convergence of the Luneburg–Kline terms is studied and is found to be quite satisfactory.