  In recent years, higher-order topological phases have attracted great
interest in various fields of physics. These phases have protected boundary
states at lower-dimensional boundaries than the conventional first-order
topological phases due to the higher-order bulk-boundary correspondence. In
this review, we summarize current research progress on higher-order topological
phases in both crystalline and non-crystalline systems. We firstly introduce
prototypical models of higher-order topological phases in crystals and their
topological characterizations. We then discuss effects of quenched disorder on
higher-order topology and demonstrate disorder-induced higher-order topological
insulators. We also review the theoretical studies on higher-order topological
insulators in amorphous systems without any crystalline symmetry and
higher-order topological phases in nonperiodic lattices including
quasicrystals, hyperbolic lattices, and fractals, which have no crystalline
counterparts. We conclude the review by a summary of experimental realizations
of higher-order topological phases and discussions on potential directions for
future study.
