The concept of band-limited fractals is introduced and used to describe the diffraction of electromagnetic and optical waves by irregular structures. This concept is demonstrated through the example of plane-wave diffraction by a fractal phase screen of finite extent. The effect of the fractal phase screen is noted on the evolution of an incident wave with the fractal dimension and other descriptors used as parameters. Of particular interest is the result that, for random fractal phase screens, the diffraction pattern from a single realization of the model phase screen can be identical to the pattern averaged over an ensemble of screens. In these cases an orderly pattern emerges from a chaotic one. This problem of fractal diffraction is of intrinsic interest because of the variety of problems found to be described by fractal, as opposed to Euclidean, geometry. The results have potential applications to the propagation of waves through random media, the reflection of waves from rough surfaces, and the characterization of these processes through remote means.