Diffusion in disordered, fractal structures is anomalous, different than that in regular space. Fractal structures are found everywhere in Nature, and as a consequence anomalous diffusion has far-reaching implications for a host of phenomena. We see its effects in flow within fractured and porous rocks, in the anomalous density of states in dilute magnetic systems, in silica aerogels and in glassy ionic conductors, anomalous relaxation in spin glasses and in macromolecules, conductivity of superionic conductors such as hollandite and of percolation clusters of Pb on thin films of Ge and Au, electron–hole recombination in amorphous semiconductors, and fusion and trapping of excitations in porous membrane films, polymeric glasses, and isotropic mixed crystals, to mention a few examples.

It was Pierre Gilles de Gennes who first realized the broad importance of anomalous diffusion, and who coined the term “the ant in the labyrinth”, describing the meandering of random walkers in percolation clusters. Since the pioneering work of de Gennes, the field has expanded very rapidly. The subject has been reviewed by several authors, including ourselves, and from various perspectives. This book builds upon our review on anomalous diffusion from 1987 and it covers the vast material that has accumulated since. Many questions that were unanswered then have been settled, yet, as usual, this has only brought forth a myriad of other questions. Whole new directions of research have emerged, most noticeably in the area of diffusion-limited reactions. The scope of developments is immense and cannot possibly be addressed in one volume. Neither do we have the necessary expertise. Hence, we have chosen once again to base the presentation mostly on heuristic scaling arguments.