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 discovered in recent studies of atomically thin two-dimensional (2D) materials.  A number of highly 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, massless excitons, tunable magnetism and plasmonic properties, electron supercollimation, novel topological phases, etc. – adding to the promise of these 2D materials for exploration of new science and valuable applications.