We consider the complete wetting transition at nonplanar wall-fluid interfaces, where the height of the substrate varies as a power-law proportional to\x\(gamma) (with exponents 0 less than or equal to gamma less than or equal to 1) in one direction (x). From a general scaling analysis, supported by numerical and analytical effective interfacial model calculations, we argue that such power-law wedges can alter the growth law describing the divergence of the interfacial height l(0) (measured from the wedge bottom) and other length scales as the bulk saturation chemical potential is approached. For realistic experimental systems with dispersion forces, we predict that the complete wetting critical exponents are determined by gamma for wedge shape with gamma > 1/2. For gamma < 1/2, the asymptotic growth of the film thickness should be similar to that found for planar systems. Nevertheless, crossover behavior due to the influence of the geometry is still observable in adsorption isotherms. (C) 2000 American Institute of Physics. [S0021-9606(00)70210-8].