This Chapter illustrates a procedure for the minimal mass design of tensegrity systems under yielding and buckling constraints. An optimization method based on an iterative linear programming algorithm is employed, drawing from the multi-faceted studies conducted by Robert Skelton and co-workers in the relevant research domain. Minimal mass designs for simply-supported and cantilever beams with tensegrity architecture are shown to exhibit a significantly lower mass, as compared to design procedures based on conventional structural shapes. Newly designed, spider-shaped tensegrity systems are examined for the case of a simply-supported beam, while a cantilever beam example employs the well-known Michell truss topology.