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Membranes composed of amphiphilic molecules are highly§flexible surfaces that determine the architecture of§biological systems and provide a basic structural element§for complex fluids such as microemulsions. Recently, a§variety of new experimental methods such as X-ray§scattering, neutron scattering, and atomic force microscopy§have been used in order to study themolecular structure of§these membranes. Their conformational behavior, on the other§hand, is studied by optical and electron microscopy, which§reveals that membranes in aqueous solution exhibit an§amazing variety of different shapes. Several theoretical§concepts are described suchas bending elasticity,§curvature, and minimal surfaces in order to understand this§polymorphism. These concepts are also useful to describe the§behavior of membranes in complex fluids where they can build§up hexagonal, lamellar, triply-periodic, cubic, and sponge§phases. The contributions to this volume provide an§up-to-date overview and describe thestate-of-the-art of§this rapidly evolving field of research.