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The function of biomembranes as tough, flexible partitions of cellular organelles involves a rich diversity of dynamic phenomena. Processes involving membrane shape transformations such as exocytosis are controlled by the specific action of molecular machinery, which transduces readily available chemical energy and uses it to overcome viscous damping and elastic energy barriers. On the micron scale, cell membranes typically inhabit an environment out of equilibrium. Their motion is governed by the interplay between thermal energy, strong dissipation due to the surrounding fluid, and chemical energy. The thermal part is not only responsible for the incessant fluctuations of bilayers, but also leads to nontrivial entropic forces.