  With the rapid development of topological states in crystals, the study of
topological states has been extended to quasicrystals in recent years. In this
review, we summarize the recent progress of topological states in
quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We
first give a brief introduction to quasicrystalline structures. Then, we
discuss topological phases in 1D quasicrystals where the topological nature is
attributed to the synthetic dimensions associated with the quasiperiodic order
of quasicrystals. We further present the generalization of various types of
crystalline topological states to 2D quasicrystals, where real-space
expressions of corresponding topological invariants are introduced due to the
lack of translational symmetry in quasicrystals. Finally, since quasicrystals
possess forbidden symmetries in crystals such as five-fold and eight-fold
rotation, we provide an overview of unique quasicrystalline symmetry-protected
topological states without crystalline counterpart.
