We discuss the structure of those polytopes in Rⁿ+ that are Minkowski sums of prisms. A prism is the convex hull of the origin and n positive multiples of the unit vectors. We characterize the defining outer surface of such polytopes by describing the shape of all maximal faces. As this shape resembles the view of a cephalopod, the polytope obtained is called a 'cephoid'. The general geometrical and combinatorial aspects of cephoids are exhibited.