Euclidean medial axes and their `parents', symmetry sets, have been used in shape recognition for some time. I shall outline some of the underlying mathematics, in 2D and 3D, and give some details of recent attempts to extend the construction to one invariant under affine transformations. An introduction to the latter (much more technical than my talk) is P.J.Giblin and G.Sapiro, `Affine-Invariant Distances, Envelopes and Symmetry Sets', Geometriae Dedicata 71 (1998), 237-261.