In addition to sheet structures with purely hexagonal carbon rings, which naturally form surfaces of zero Gaussian curvature such as sheets and tubes, a graphenic membrane can also assume a conical shape whose apex is defined by one or more disclinations taking the form of fivefold (or possibly smaller) rings. Geometrically, just as a sheet of paper with a wedge removed can be resealed to form a conical hat, a graphene sheet with a wedge removed (i.e., a disclination) can be resealed, notionally, to form a cone or horn. The single-wall carbon nanohorns (SWNH) form one class of such conical structures, with a particularly sharp apical angle, a well-characterized high-yield synthesis route, and a distinct aggregate microstructure. Conical graphenic structures with wider opening angles, corresponding to fewer pentagonal disclinations at the apex, also form, sometimes as multilayered structures. The pentagonal defects in carbon nanocones perturb the low-energy electronic structure both locally and globally, defining both a local region of enhanced reactivity and a global geometric phase relation with profound consequences for electron transport around the apex. The rapid variation in local sheet orientation around the cone and the two-dimensional nature of the electronic states within imply that uniform laboratory fields can generate highly nonuniform effective local fields for states in the cone.