  We propose a general principle for constructing higher-order topological
(HOT) phases. We argue that if a $D$-dimensional first-order or regular
topological phase involves $m$ Hermitian matrices that anti-commute with
additional $p-1$ mutually anti-commuting matrices, it is conceivable to realize
an $n$th-order HOT phase, where $n=1, \cdots, p$, with appropriate combinations
of discrete symmetry-breaking Wilsonian masses. An $n$th-order HOT phase
accommodates zero modes on a surface with codimension $n$. We exemplify these
scenarios for prototypical three-dimensional gapless systems, such as a
nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac
semimetals, transforming under (pseudo-)spin-$\frac{1}{2}$ or 1
representations. The former system permits an unprecedented realization of a
fourth-order phase, without any surface zero modes. Our construction can be
generalized to HOT insulators and superconductors in any dimension and symmetry
class.
