The propagation of stationary three-dimensional nonlinear guided waves and of Gaussian beams in a slab waveguide in which the cladding displays both Kerr-law and saturable instantaneous self-focusing nonlinearities is simulated numerically. The numerical analysis supports the hypothesis that the stationary three-dimensional nonlinear guided waves are stable on propagation for propagation constants corresponding to the ascending branches of the nonlinear dispersion curves. The numerical simulations of the Gaussian-beam propagation in the nonlinear slab waveguide illustrate both diffraction and nonlinear effects such as solitary-wave emission. The transparent-boundary condition, which employs no adjustable parameters and is easily incorporated into a Crank–Nicolson finite-difference scheme, was used in the three-dimensional calculations.