The Horrocks bundle (not to be confused with the Horrocks-Mumford bundle) is an indecomposable holomorphic rank 3 bundle over CP^5 with c_2=3. It can be interpreted as a solution of the Yang-Mills equations on the quaternionic projective plane HP^2, via a Penrose transform and twistor equation. In joint work with Udhav Fowdar, this realization leads to the existence of nowhere-vanishing spinors on HP^2 invariant by SU(3). I shall use this example to introduce instantons associated with G-structures in higher dimensions, cohomogeneity-one actions by SU(3), and Spin(7) structures on 8-manifolds.