Eight structural elements in biological materials are identified as the most common amongst a variety of animal taxa. These are proposed as a new paradigm in the field of biological materials science as they can serve as a toolbox for rationalizing the complex mechanical behavior of structural biological materials and for systematizing the development of bioinspired designs for structural applications. They are employed to improve the mechanical properties, namely strength, wear resistance, stiffness, flexibility, fracture toughness, and energy absorption of different biological materials for a variety of functions (e.g., body support, joint movement, impact protection, weight reduction). The structural elements identified are: fibrous, helical, gradient, layered, tubular, cellular, suture, and overlapping. For each of the structural design elements, critical design parameters are presented along with constitutive equations with a focus on mechanical properties. Additionally, example organisms from varying biological classes are presented for each case to display the wide variety of environments where each of these elements is present. Examples of current bioinspired materials are also introduced for each element.