  Carbon has numerous one-dimensional (1D), two-dimensional (2D), and
three-dimensional (3D) allotropic structures. The study of carbon materials has
been a major focus of material science and condensed matter physics. Previous
studies have identified different classes of topological semimetallic carbon
allotropes with different topological phases. In this review, we first give a
brief summary of the development of carbon allotropes from 1D to 3D. Next, we
discuss topological properties of carbon materials and their physical origin.
Then, we consider possible expansion of the topological study of carbon
materials to other light-element materials such as boron. Finally, we present
future prospects in pursue of topological physics within carbon allotropes.
