Despite centuries of intense effort amongst mathematicians, and a huge literature in this field, there has previously never been a generally valid mathematical model of ultrathin elastic «shells» (actually surfaces) of revolution. This survey paper presents the first theoretical framework capable of unifying, into a single coherent body of knowledge, a diversity of shapes associated with elastic bows, car bumper-bars, molluscan shells, even flower-buds and pine-cones. It becomes apparent why conventional analysis enjoys limited success when approximating elastic cones to perturbedcylinders anddiscs. Also the paper provides a theoretical basis for analysing the wrinkling of compressed engineering structures. These successes, the new unification and the simplicity of relevant theory which maynever in principle be capable of working in this context.