The regions of validity of the Kirchhoff and the first-order small-perturbation models in rough surface scattering are examined by numerical simulation. The procedure is to generate a one dimensional perfectly conducting random surface on the digital computer, compute the induced current on the surface due to an impinging plane wave by the moment method, and then calculate the far zone backscattered field and power. This procedure is repeated at least forty times and the results averaged to obtain the average scattered power. It is important to note that many so called Kirchhoff models in the literature involve simplifying assumptions in addition to the Kirchhoff approximation. To avoid confusion the Kirchhoff model in this study uses only the Kirchhoff approximation for the surface current. The far zone scattered field and power are evaluated numerically without further simplifying assumptions. Comparisons between the numerical calculations and the models are made for various values of the surface rms height and correlation length both normalized to the incident wave number (denoted by kσ and kL, respectively). By using these two parameters to form a two-dimensional space the approximate regions of validity are then established. It is found that due to the inclusion of the coherent scattering component the Kirchhoff model continues to provide good agreement with numerical calculations over small angles of incidence when kσ is less than 0.2 and kL < 2.0, a region where the Kirchhoff method was expected to fail. It is also found that the usual requirements of small height and small slope are inadequate to guarantee the validity of the first-order perturbation model. It is necessary that kL also be small.