This article reports some recent progress on elasticities of fluid membranes and polymer membranes. The shape equation of lipid vesicles, the shape equation and boundary conditions of open lipid bilayers and vesicles with lipid domains are presented. The free energy of a membrane consisting of polymers is derived by scaling analysis. The first-order variation of the free energy gives the shape equation and in-plane strain equations of polymer membranes, and the second-order variation arrives at mechanical stabilities of polymer membranes. (c) 2005 Elsevier B.V. All rights reserved.