Materials with a cellular structure are widespread in nature and include wood, cork, plant parenchyma and trabecular bone. Natural cellular materials are often mechanically efficient:​ the honeycomb-like microstructure of wood, for instance, gives it an exceptionally high performance index for resisting bending and buckling. Here we review the mechanics of a wide range of natural cellular materials and examine their role in lightweight natural sandwich structures (e.g. iris leaves) and natural tubular structures (e.g. plant stems or animal quills). We also describe two examples of engineered biomaterials with a cellular structure, designed to replace or regenerate tissue in the body.