Using density functional calculations, we demonstrate that the planarity of the nonclassical planar tetracoordinate carbon (ptC) arrangement can be utilized to construct new families of flat, tubular, and cage molecules which are geometrically akin to graphenes, carbon nanotubes, and fullerenes but have fundamentally different chemical bonds. These molecules are assembled with a single type of hexagonal blocks called starbenzene (D(6h) C(6)Be(6)H(6)) through hydrogen-bridge bonds that have an average bonding energy of 25.4-33.1 kcal mol(-1). Starbenzene is an aromatic molecule with six pi electrons, but its carbon atoms prefer ptC arrangements rather than the planar trigonal sp(2) arrangements like those in benzene. Various stability assessments indicate their excellent stabilities for experimental realization. For example, one starbenzene unit in an infinite two-dimensional molecular sheet lies on average 154.1 kcal mol(-1) below three isolated linear C(2)Be(2)H(2) (global minimum) monomers. This value is close to the energy lowering of 157.4 kcal mol(-1) of benzene relative to three acetylene molecules. The ptC bonding in starbenzene can be extended to give new series of starlike monocyclic aromatic molecules (D(4h) C(4)Be(4)H(4)(2-), D(5h) C(5)Be(5)H(5)(-), D(6h) C(6)Be(6)H(6), D(7h) C(7)Be(7)H(7)(+), D(8h) C(8)Be(8)H(8)(2-), and D(9h) C(9)Be(9)H(9)(-)), known as starenes. The starene isomers with classical trigonal carbon sp(2) bonding are all less stable than the corresponding starlike starenes. Similarly, lithiated C(5)Be(5)H(5) can be assembled into a C(60)-like molecule. The chemical bonding involved in the title molecules includes aromaticity, ptC arrangements, hydrogen-bridge bonds, ionic bonds, and covalent bonds, which, along with their unique geometric features, may result in new applications.