Carbon allotropes comprised exclusively of sp² hybridized carbon atoms present flat or curved surface, whose overall geometric feature is reflected by its curvature. Consisting of six-membered rings exclusively, graphene has zero curvature. Five-membered rings induce positive curvature as displayed in fullerenes; while seven- or eight-membered rings in a hexagonal lattice of carbon induce negative curvature as displayed in a saddle-shaped surface. Negatively curved nanocarbons of three-dimensional periodic structures are named as Mackay crystal or carbon schwarzites, which are a long-sought target in carbon nanoscience, but have not been synthesized unambiguously yet. A bottom-up approach to negatively curved nanocarbons involves synthesis of negatively curved polycyclic arenes, which can then be used as templates or monomers in synthesis of larger nanocarbons. Negatively curved polycyclic arenes, which can be designed and synthesized by introduction of seven or eight-membered rings into polycyclic aromatic frameworks, exhibit structural characteristics and properties that are not available to planar polycyclic arenes. Taking the polycyclic arenes containing eight-membered rings as examples, this review article describes research in the design, synthesis, stereochemical dynamics, and other characteristics of negatively curved polycyclic arenes, and provides an outlook for new directions in the research of negatively curved nanocarbons.