In solids, electrons move quantum mechanically and experience correlations with other electrons and ions due to the Coulomb interactions. The quantum nature and correlations of electrons lead to surprising “quantum-enabled” phenomena, namely topological phases. In this colloquium, we will discuss the physics of such topological states and review the recent progress of the research field. In particular, when the topological states are intertwined with geometric backgrounds (for example, crystalline symmetries), exotic fractional excitations such as Majorana fermion and anyon can emerge. We theoretically demonstrate when such fractional excitations can emerge and how they can be manifested.