W. Helfrich proposed a theory of the elasticity of lipid bilayers, distinguishing three types of strain: stretching, tilt, and curvature. The curvature elasticity is the primary factor controlling the shape of nonspherical vesicles. The theory derives Euler-Lagrange equations for the shape of vesicles in magnetic fields and under excess external pressure, showing that magnetic fields can deform spherical vesicles into ellipsoids of revolution, while excess pressure can destabilize the spherical shape. Spontaneous curvature of the bilayer influences these effects. The article discusses possible experiments to determine the elastic properties of lipid bilayers, including measuring the curvature-elastic modulus and spontaneous curvature. The elasticity of lipid bilayers is analyzed through stretching, tilt, and curvature strains. Stretching involves changes in area due to tension, with the elastic energy being a quadratic function of the relative area change. Tilt involves changes in the orientation of lipid molecules, with the elastic energy depending on the tilt angle. Curvature elasticity is described by the curvature of the bilayer, with the energy depending on the derivatives of the layer normal. The curvature-elastic energy is expressed in terms of the layer normal and its derivatives, and the theory shows that curvature is the dominant factor in determining the shape of nonspherical vesicles. The article also discusses the effects of magnetic fields and pressure on vesicle shapes, showing that magnetic fields can induce ellipsoidal deformations, while pressure can lead to instability. The theory is applied to derive the shape of rotationally symmetric vesicles, considering the curvature-elastic energy and the effects of magnetic fields and pressure. The results show that the shape of vesicles can be influenced by these factors, with the curvature-elastic energy playing a key role in determining the equilibrium shape. The article concludes with a discussion of the implications of the theory for understanding the behavior of lipid bilayers and vesicles, highlighting the importance of curvature elasticity in controlling their shapes and the potential for experimental verification of the theoretical predictions.W. Helfrich proposed a theory of the elasticity of lipid bilayers, distinguishing three types of strain: stretching, tilt, and curvature. The curvature elasticity is the primary factor controlling the shape of nonspherical vesicles. The theory derives Euler-Lagrange equations for the shape of vesicles in magnetic fields and under excess external pressure, showing that magnetic fields can deform spherical vesicles into ellipsoids of revolution, while excess pressure can destabilize the spherical shape. Spontaneous curvature of the bilayer influences these effects. The article discusses possible experiments to determine the elastic properties of lipid bilayers, including measuring the curvature-elastic modulus and spontaneous curvature. The elasticity of lipid bilayers is analyzed through stretching, tilt, and curvature strains. Stretching involves changes in area due to tension, with the elastic energy being a quadratic function of the relative area change. Tilt involves changes in the orientation of lipid molecules, with the elastic energy depending on the tilt angle. Curvature elasticity is described by the curvature of the bilayer, with the energy depending on the derivatives of the layer normal. The curvature-elastic energy is expressed in terms of the layer normal and its derivatives, and the theory shows that curvature is the dominant factor in determining the shape of nonspherical vesicles. The article also discusses the effects of magnetic fields and pressure on vesicle shapes, showing that magnetic fields can induce ellipsoidal deformations, while pressure can lead to instability. The theory is applied to derive the shape of rotationally symmetric vesicles, considering the curvature-elastic energy and the effects of magnetic fields and pressure. The results show that the shape of vesicles can be influenced by these factors, with the curvature-elastic energy playing a key role in determining the equilibrium shape. The article concludes with a discussion of the implications of the theory for understanding the behavior of lipid bilayers and vesicles, highlighting the importance of curvature elasticity in controlling their shapes and the potential for experimental verification of the theoretical predictions.