Faced with a comparatively limited palette of minerals and organic polymers as building materials, evolution has arrived repeatedly on structural solutions that rely on clever geometric arrangements to avoid mechanical trade-offs in stiffness, strength and flexibility. In this tutorial review, we highlight the concept of tessellation, a structural motif that involves periodic soft and hard elements arranged in series and that appears in a vast array of invertebrate and vertebrate animal biomaterials. We start from basic mechanics principles on the effects of material heterogeneities in hypothetical structures, to derive common concepts from a diversity of natural examples of one-, two- and three-dimensional tilings/layerings. We show that the tessellation of a hard, continuous surface - its atomization into discrete elements connected by a softer phase - can theoretically result in maximization of material toughness, with little expense to stiffness or strength. Moreover, the arrangement of soft/flexible and hard/stiff elements into particular geometries can permit surprising functions, such as signal filtering or 'stretch and catch' responses, where the constrained flexibility of systems allows a built-in safety mechanism for ensuring that both compressive and tensile loads are managed well. Our analysis unites examples ranging from exoskeletal materials (fish scales, arthropod cuticle, turtle shell) to endoskeletal materials (bone, shark cartilage, sponge spicules) to attachment devices (mussel byssal threads), from both invertebrate and vertebrate animals, while spotlighting success and potential for bio-inspired manmade applications.