  In this tutorial, we pedagogically review recent developments in the field of
non-interacting fermionic phases of matter, focussing on the low energy
description of higher-order topological insulators in terms of the Dirac
equation. Our aim is to give a mostly self-contained treatment. After
introducing the Dirac approximation of topological crystalline band structures,
we use it to derive the anomalous end and corner states of first- and
higher-order topological insulators in one and two spatial dimensions. In
particular, we recast the classical derivation of domain wall bound states of
the Su-Schrieffer-Heeger (SSH) chain in terms of crystalline symmetry. The edge
of a two-dimensional higher-order topological insulators can then be viewed as
a single crystalline symmetry-protected SSH chain, whose domain wall bound
states become the corner states. We never explicitly solve for the full
symmetric boundary of the two-dimensional system, but instead argue by
adiabatic continuity. Our approach captures all salient features of
higher-order topology while remaining analytically tractable.
