The state-of-the-art in theoretical research directed at the search for superhard materials based on carbon fullerenes and nanotubes has been briefly reviewed. The data available on elastic properties of the main groups of condensed phases that form due to the interaction of fullerenes or (and) nanotubes because of both weak (of the van der Waals type) bonds and polymerization to form strong covalent sp3 bonds have been considered. Models for new hypothetical carbon nanostructured materials (the so-called autointercalated hyperdiamonds, covalently bonded lattices of nanotubes, tubular cubic crystals, etc.) are discussed, and elastic properties of them are considered as related to their atomic structure, special features of their electronic structure and chemical bond have been discussed. The problems of synthesis of these systems are briefly considered.