The four real division algebras (reals, complexes, quarternions and octonions) are signposts to select mathematical structures. Using the tool of adjoint division algebras, with respect to which the division algebras themselves appear in the role of spinor spaces, some of these structures are developed, including parallelizable spheres, exceptional Lie groups, and triality. In the case of triality the use of adjoint octonions simplifies its investigation. Motivating this work, however, is a strong conviction that the design of our physical reality arises from this select mathematical realm. A case for that conviction is presented; a derivation of the standard model of leptons and quarks. The book aims to be of particular interest to particle and high energy theorists, and to applied mathematicians.