  The surrounding world surprises us by the beauty and variety of complex
shapes that emerge from nanometric to macroscopic scales. Natural or
manufactured materials (sandstones, sedimentary rocks and cement), colloidal
solutions (proteins and DNA), biological cells, tissues and organs (lungs,
kidneys and placenta), they all present irregularly shaped "scenes" for a
fundamental transport "performance", that is, diffusion. Here, the geometrical
complexity, entangled with the stochastic character of diffusive motion,
results in numerous fascinating and sometimes unexpected effects like diffusion
screening or localization. These effects control many diffusion-mediated
processes that play an important role in heterogeneous catalysis, biochemical
mechanisms, electrochemistry, growth phenomena, oil recovery, or building
industry. In spite of a long and rich history of academic and industrial
research in this field, it is striking to see how little we know about
diffusion in complex geometries, especially the one which occurs in three
dimensions.
  We present our recent results on restricted diffusion. We look into the role
of geometrical complexity at different levels, from boundary microroughness to
hierarchical structure and connectivity of the whole diffusion-confining
domain. We develop a new approach which consists in combining fast random walk
algorithms with spectral tools. The main focus is on studying diffusion in
model complex geometries (von Koch boundaries, Kitaoka acinus, etc.), as well
as on developing and testing spectral methods. We aim at extending this
knowledge and at applying the accomplished arsenal of theoretical and numerical
tools to structures found in nature and industry.
