  We are concerned with the problem of the stability of the syzygy bundles
associated to base point free vector spaces of forms of the same degree d on
the projective space of dimension n. We deduce directly, from Mark Green's
vanishing theorem for Koszul cohomology, that any such bundle is stable if his
rank is sufficiently high. With a similar argument, we prove the semistability
of a certain syzygy bundle on a general complete intersection of hypersurfaces
of degree d in the projective space. This answers a question of H. Flenner
(1984). We then give an elementary proof of H. Brenner's criterion of stability
for monomial syzygy bundles, avoiding the use of Klyachko's results on toric
vector bundles. We finally prove the existence of stable syzygy bundles defined
by monomials of the same degree d, of any possible rank, for n at least 3. This
extends the similar result proved, for n=2, by L. Costa, P. Macias Marques and
R.M. Miro-Roig (2009). The extension to the case n at least 3 has been also,
independently, obtained by P. Macias Marques in his thesis (2009).
