  We study the Rarita-Schwinger operator on compact Riemannian spin manifolds.
In particular, we find examples of compact Einstein manifolds with positive
scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel.
For positive quaternion K\"ahler manifolds and symmetric spaces with spin
structure we give a complete classification of manifolds admitting
Rarita-Schwinger fields. In the case of Calabi-Yau, hyperk\"ahler, $G_2$ and
Spin(7) manifolds we find an identification of the kernel of the
Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a
classification of compact irreducible spin manifolds admitting parallel
Rarita-Schwinger fields.
