From our investigations of natural composite materials such as abalone shell and bone we have learned the following. (1) Nature is frugal with resources: it uses just a few per cent glue, by weight, to glue together composite materials. (2) Nature does not avoid voids. (3) Nature makes optimized glues with sacrificial bonds and hidden length. We discuss how optimized adhesives combined with high specific stiffness/strength structures such as carbon nanotubes or graphene sheets could yield remarkably strong, lightweight, and damage-resistant materials.