In this talk, we present some new physical phenomena found in recent theoretical studies of atomically thin two-dimensional (2D) materials. Because of reduced dimensionality, interaction and symmetry effects as well as environmental screening effects dominate many properties of these systems, leading to manifestation of concepts and phenomena that may not be so prominent or have not been seen in bulk materials. A number of highly interesting and unexpected phenomena have been discovered: strongly bound excitons with unusual energy level structures and optical selection rules; exchange-induced light-like (massless) exciton dispersion; tunable optical and plasmonic properties; electron supercollimation by 1D disorder in graphene and related 2D Dirac materials; and novel topological phases in graphene nanoribbons. We describe their physical origin and compare theoretical predictions with experimental results. This work was supported in part by the National Science Foundation, the U.S. Department of Energy and the Office of Naval Research.