  A Dirac bundle is a euclidean bundle over a riemannian manifold $M$ which is
a compatible left $C\ell(M)$-module, together with a metric connection also
compatible with the Clifford action in a natural way. We prove some vanishing
theorems and introduce the twistor equation within this framework. In
particular, we exhibit a characterization of solutions for this equation in
terms of the Dirac operator $D$ and a suitable Weitzenb\"ock-type curvature
operator $\mathcal{R}$. Finally, we analyze the especial case of the Clifford
bundle to prove existence of nontrivial solutions of the twistor equation on
spheres.
