Contains a comprehensive description of the mechanical equilibrium and deformation of membranes as a surface problem in differential geometry. Following the pioneering work by W. Helfrich, the fluid membrane is seen as a nematic or smectic - a liquid crystal film and its elastic enegy form is deduced exactly from the curvature elastic theory of the liquid crystals. With surface variation, the minimization of the energy at the fixed osmotical pressure and surface tension gives a surface equation in geometry that involves potential interest in mathematics. This text presents the results of the...