  We devise a generic recipe for constructing $D$-dimensional lattice models
whose $d$-dimensional boundary states, located on surfaces, hinges, corners,
and so forth, can be obtained exactly. The solvability is rooted in the
underlying lattice structure and as such does not depend on fine tuning,
allowing us to track their evolution throughout various phases and across phase
transitions. Most saliently, our models provide "boundary solvable" examples of
the recently introduced higher-order topological phases. We apply our general
approach to breathing and anisotropic kagome and pyrochlore lattices for which
we obtain exact corner eigenstates, and to periodically driven two-dimensional
models as well as to three-dimensional lattices where we present exact
solutions corresponding to one-dimensional chiral states at the hinges of the
lattice. We relate the higher-order topological nature of these models to
reflection symmetries in combination with their provenance from
lower-dimensional conventional topological phases.
