In this mini-review, a brief historical survey of the mechanisms which determine the shapes of liposomes and cells and the budding and fission of their membrane is presented. Special attention is given to the role of orientational ordering of membrane components in thin membrane necks which connect the membrane buds (daughter vesicles) to the parent membrane. It is indicated that topological anti-defects in membrane necks may induce the rupture of the neck and the fission of the membrane daughter vesicles.
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