Biological membranes separate cells and organelles from their surrounding environment thus defining their identity. They maintain different chemical environments within and outside the cell while allowing continuous transport of nutrients into the cell. These surfaces are made of lipid molecules that exhibit a unique balance between love and fear of water that grants them their novel physical behavior. They are neither solid nor fluid – a property that allows them to maintain their structure yet undergo large morphological transformations enabling the cells to grow, move and divide. Remarkably, while engineers have historically avoided instabilities from structural response by careful designing, instabilities play a critical role in interfacial phenomena in cells. In this talk, I will present two examples to argue my case. The first example pertains to the morphogenesis of the nuclear envelope. Here, nanoscale buckling instabilities during the growth phase impart the nuclear envelope its fascinating ultra-donut topology. The second example concerns the fission of mitochondria. Here, lipid-catalyzed geometric instabilities catalyze fission and help achieve stable superconstrictions during extreme structural remodeling.