Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which are characterized by a Cayley structure. Examples are projective duality of toric varieties and polyhedral adjunction theory.