  Higher order topological insulators are a new class of topological insulators
in dimensions $\rm d>1$. These higher-order topological insulators possess $\rm
(d - 1)$-dimensional boundaries that, unlike those of conventional topological
insulators, do not conduct via gapless states but instead are themselves
topological insulators. Precisely, an $\rm n^{\rm th}$-order topological
insulator in $\rm m$ dimensions hosts $\rm d_{c} = (m - n)$-dimensional
boundary modes $\rm (n \leq m)$. For instance, a three-dimensional second
(third) order topological insulator hosts gapless modes on the hinges
(corners), characterized by $\rm d_{c} = 1 (0)$. Similarly, a second order
topological insulator in two dimensions only has gapless corner states ($\rm
d_{c} = 0$) localized at the boundary. These higher order phases are protected
by various crystalline symmetries. Moreover, in presence of proximity induced
superconductivity and appropriate symmetry breaking perturbations, the above
mentioned bulk-boundary correspondence can be extended to higher order
topological superconductors hosting Majorana hinge or corner modes. Such
higher-order systems constitute a distinctive new family of topological phases
of matter which has been experimentally observed in acoustic systems,
multilayer $\rm WTe_{2}$ and $\rm Bi_{4}Br_{4}$ chains. In this general
article, the basic phenomenology of higher order topological insulators and
higher order topological superconductors are presented along with some of their
experimental realization.
