  It is a longstanding problem in Algebraic Geometry to determine whether the
syzygy bundle $E_{d_1,..., d_n}$ on $\PP^N$ defined as the kernel of a general
epimorphism $\xymatrix{\phi:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ is
(semi)stable. In this note, we restrict our attention to the case of syzygy
bundles $E_{d,n}$ on $\PP^N$ associated to $n$ generic forms $f_1,...,f_n\in
K[X_0,X_1,..., X_N]$ of the same degree $d$. Our first goal is to prove that
$E_{d,n}$ is stable if $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves,
in general, the bound $n\le d(N+1)$ given by G. Hein in \cite{B}, Appendix A.
  In the last part of the paper, we study moduli spaces of stable rank $n-1$
vector bundles on $\PP^N$ containing syzygy bundles. We prove that if $N+1\le
n\le\tbinom{d+2}{2}+N-2$ and $N\ne 3$, then the syzygy bundle $E_{d,n}$ is
unobstructed and it belongs to a generically smooth irreducible component of
dimension $n\tbinom{d+N}{N}-n^2$, if $N \geq 4$, and
$n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2$, if N=2.
