  We formulate and study the negative gradient flow of an energy functional of
Spin(7)-structures on compact 8-manifolds. The energy functional is the
$L^2$-norm of the torsion of the Spin(7)-structure. Our main result is the
short-time existence and uniqueness of solutions to the flow. We also explain
how this negative gradient flow is the most general flow of Spin(7)-structures.
We also study solitons of the flow and prove a non-existence result for compact
expanding solitons.
