Is shape just geometric similarity? Surely not; on the other hand, the main topological type of equivalence, i.e., homeomorphism, is much too far from the human idea of "having the same shape". Still, this does not mean that we have to avoid mathematical tools in shape description, analysis, comparison, classification, retrieval… We start from the work of Jan Koenderink, seeing how differential topology yields important cues for understanding shape, and we end up with ideas and applications of persistent homology.