Three-dimensional molecular structures of topologically nontrivial fullerenes (consisting of either finite or extended structures) are usually aesthetically pleasing. In this article, we demonstrate that beads such as the ones commonly used in decorative art and ornament making can also be used to construct arbitrary fullerene structures. Based on the spiral codes of fullerenes, we developed a systematic strategy for making physical models of cage-like fullerenes use common beads. The resulting beaded model structure is similar to the true three-dimensional molecular structure of corresponding fullerene due to an interesting analogy between the hard-sphere repulsion among neighboring beads and the microscopic valence shell electron pair repulsion for the sp2-hybridized carbon atoms. More complicated fullerenes models that have nontrivial topology (e.g. toroidal carbon nanotubes, helically coiled carbon nanotubes, and high-genus fullerenes) can also be faithfully constructed using beads. Beaded models of extended graphitic structures such as those that correspond to tiling of graphene sheet on a Schwartz P- and D- surfaces, Shoen I-WP, and Nervious surfaces, can also been created.