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In recent years, higher-order topological phases have attracted greatinterest in various fields of physics. These phases have protected boundarystates at lower-dimensional boundaries than the conventional first-ordertopological phases due to the higher-order bulk-boundary correspondence. Inthis review, we summarize current research progress on higher-order topologicalphases in both crystalline and non-crystalline systems. We firstly introduceprototypical models of higher-order topological phases in crystals and theirtopological characterizations. We then discuss effects of quenched disorder onhigher-order topology and demonstrate disorder-induced higher-order topologicalinsulators. We also review the theoretical studies on higher-order topologicalinsulators in amorphous systems without any crystalline symmetry andhigher-order topological phases in nonperiodic lattices includingquasicrystals, hyperbolic lattices, and fractals, which have no crystallinecounterparts. We conclude the review by a summary of experimental realizationsof higher-order topological phases and discussions on potential directions forfuture study.

Higher-order topological phases in crystalline and non-crystalline systems: a review