  We study the holomorphic vector bundles E over the twistor space Tw(M) of a
compact simply connected hyperk\"ahler manifold $M$. We give a characterization
of the semistability condition for E in terms of its restrictions to the
holomorphic sections of the holomorphic twistor projection \pi
:Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection,
then E is holomorphically trivial and the holomorphic connection on E is
trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we
prove that its restriction to the generic fibre of \pi is stable. On the other
hand, for a K3 surface M, we construct examples of irreducible vector bundles
of any composite rank on Tw(M) whose restriction to every fibre of \pi is
non-stable. We have obtained a new method of constructing irreducible vector
bundles on hyperk\"ahler twistor spaces; this method is employed in
constructing these examples.
