  We formulate and study the isometric flow of $\mathrm{Spin}(7)$-structures on
compact $8$-manifolds, as an instance of the harmonic flow of geometric
structures. Starting from a general perspective, we establish Shi-type
estimates and a correspondence between harmonic solitons and self-similar
solutions for arbitrary isometric flows of $H$-structures. We then specialise
to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for
long-time existence, via a monotonicity formula along the flow, which actually
leads to an $\varepsilon$-regularity theorem. Moreover, we prove
Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the
harmonic flow, and we characterise Type-$\mathrm{I}$ singularities as being
modelled on shrinking solitons.We also establish a Bryant-type description of
isometric $\mathrm{Spin}(7)$-structures, based on squares of spinors, which may
be of independent interest.
