Due to complexity of its chemical makeup, the membrane of a biological cell has a number of interesting physical properties that other materials or material forms do not have, which also enable many biological functions of a cell. Specifically, in this work we consider a type of elasticity specific to membranes’ phospholipid bilayer which manifests in a number of potential cell shapes. Based on the experimentally obtained shapes (prolate, oblate, discocyte, stomatocyte, ...) and the data on the known shapes, we develop a theory of elasticity of closed biological membranes from the first principles of mechanics and differential geometry. Having defined the structure, we introduce Helfrich’s theory of the almost planar membrane (1973) and use it to solve the problem of a closed membrane. Given the relation for the free energy of a closed membrane, we develop a two-dimensional phase diagram of the membrane shapes. An agreement is observed between theoretically and experimentally found border lines between membrane shapes. Although it is based on basic concepts of elasticity of a regular physical membrane, the presented theory of elasticity of a closed biological membrane explains the observed shapes of biological membranes in a surprisingly faithful manner.