Symmetry, interaction and topological effects, as well as environmental screening, dominate many of the quantum properties of reduced-dimensional systems and nanostructures. These effects often lead to manifestation of counter-intuitive concepts and phenomena that may not be so prominent or have not been seen in bulk materials. In this talk, I present some fascinating physical phenomena we discovered in recent studies of atomically thin one-dimensional (1D) and two-dimensional (2D) materials. A number of interesting and unexpected behaviors have been found - e.g., strongly bound excitons (electron-hole pairs) with unusual energy level structures and new topology-dictated optical selection rules; tunable magnetism and plasmonic properties; novel topological phases; correlated multi-particle excitations; etc. - adding to the promise of 1D and 2D materials for exploration of new science and valuable applications.