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The surrounding world surprises us by the beauty and variety of complexshapes that emerge from nanometric to macroscopic scales. Natural ormanufactured materials (sandstones, sedimentary rocks and cement), colloidalsolutions (proteins and DNA), biological cells, tissues and organs (lungs,kidneys and placenta), they all present irregularly shaped "scenes" for afundamental transport "performance", that is, diffusion. Here, the geometricalcomplexity, entangled with the stochastic character of diffusive motion,results in numerous fascinating and sometimes unexpected effects like diffusionscreening or localization. These effects control many diffusion-mediatedprocesses that play an important role in heterogeneous catalysis, biochemicalmechanisms, electrochemistry, growth phenomena, oil recovery, or buildingindustry. In spite of a long and rich history of academic and industrialresearch in this field, it is striking to see how little we know aboutdiffusion in complex geometries, especially the one which occurs in threedimensions.  We present our recent results on restricted diffusion. We look into the roleof geometrical complexity at different levels, from boundary microroughness tohierarchical structure and connectivity of the whole diffusion-confiningdomain. We develop a new approach which consists in combining fast random walkalgorithms with spectral tools. The main focus is on studying diffusion inmodel complex geometries (von Koch boundaries, Kitaoka acinus, etc.), as wellas on developing and testing spectral methods. We aim at extending thisknowledge and at applying the accomplished arsenal of theoretical and numericaltools to structures found in nature and industry.

Surveying Diffusion in Complex Geometries. An Essay