We consider the problem of a thin layer of a power-law liquid falling down an inclined plate. The corresponding nonlinear evolution equation for the film thickness is solved numerically in a periodic domain. Numerical calculations show that saturation of nonlinear interactions occurs, resulting in a finite-amplitude permanent wave. The free-surface evolution is similar to that for a Newtonian liquid, but the shape and amplitude of the permanent wave are influenced strongly by the non-Newtonian fluid behaviour. The permanent wave is nearly harmonic for initial wavenumbers just below the cut-off wavenumber or of solitary type for much smaller wavenumbers. For shear-thickening (shear-thinning) liquids. the maximum wave amplitude is always smaller (larger) than that for a Newtonian liquid. For shear-thinning liquids at low flow rates, the permanent wave remains harmonic as for a Newtonian liquid. The increase of the Reynolds number, which measures the flow rate, leads to more significant growth of the two-dimensional disturbances in the case of a shear-thinning liquid than in the case of a Newtonian liquid. (C) 2004 Elsevier B.V. All rights reserved.