Carbon-rich organic compounds are candidates for use in a wide range of new technologies. Particularly interesting are materials with a framework of pure carbon, namely the allotropes. Much is known about the most common allotropes, diamond (sp 3 -carbon) and graphite (sp 2 -carbon), as well as the newer allotropes, fullerenes, nanotubes, and graphene. On the other hand, little is known about the allotrope constructed from sp-hybridized carbon atoms, a material commonly called carbyne. Without a defined sample of carbyne to study, synthetic chemists have attempted to model the properties of carbyne through the formation of defined length oligomers based on a cumulene or polyyne skeleton. Spectroscopic analysis of these series of oligomers by vibrational (IR and Raman), NMR, and UV/Vis spectroscopy has outlined some of the potential properties of carbyne. Likewise, X-ray crystallographic analyses give further clues as to the structure of carbyne. This review will highlight selected synthetic methods for cumulenes and polyynes, and it then summarizes characterization data related to structure and bonding in these molecules.