The 2016 Nobel Prizes were awarded to Yoshinori Ohsumi for autophagy and to David Thouless, Duncan Haldane and Michael Kosterlitz for topological transitions. Both of these phenomena are intrinsically related when it comes to membranes. Here, we give a brief account on topological transformations of lipid membranes, commonly known as membrane fusion and membrane scission, and introduce the underlying topological invariant, the genus. The genus of a shape is a useful concept to distinguish unambiguously the processes of membrane fusion/scission and offers a simple method to describe complex, cellular membrane structures, such as fenestrated cristae. We distinguish and highlight the connection between topological transformations of lipid membranes and the recent awards, and point out the extraordinarily large number of topological changes during autophagy.