Diffusive transport of heat, mass, and momentum across walls with irregular and fractal geometry is discussed. A configuration is considered in detail, in which the transported variable diffuses across a two-dimensional periodic irregular wall toward an overlying plane wall, while the value of the variable over each wall is maintained at a constant value. Several families of self-similar wall geometries with increasingly finer structure, eventually leading to fractal shapes, are considered in detail using an efficient numerical method that is based on conformal mapping. The numerical procedure involves the iterative solution of a large system of nonlinear algebraic equations. Computed patterns of iso-scalar contours reveal the precise effect of the shape, size, and total length of boundary irregularities on the local and total transport rates, and illustrate the enhancement in transport efficacy with wall refinement. The total rate of transport across walls with self-similar irregularities is shown to be remarkably close to that across walls with random irregularities of same roughness height.