  It is a longstanding problem in Algebraic Geometry to determine whether the
syzygy bundle $E_{d_1,...,d_n}$ on $\mathbb{P}^N$ defined as the kernel of a
general epimorphism \[\phi:\mathcal{O}(-d_1)\oplus...\oplus\mathcal{O}(-d_n)
\to\mathcal{O}\] is (semi)stable. In this thesis, attention is restricted to
the case of syzygy bundles $\mathrm{Syz}(f_1,...,f_n)$ on $\mathbb{P}^N$
associated to $n$ generic forms $f_1,...,f_n\in K[X_0,...,X_N]$ of the same
degree $d$, for ${N\ge2}$.
  The first goal is to prove that $\mathrm{Syz}(f_1,...,f_n)$ is stable if
\[N+1\le n\le\tbinom{d+N}{N},\] except for the case ${(N,n,d)=(2,5,2)}$. The
second is to study moduli spaces of stable rank ${n-1}$ vector bundles on
$\mathbb{P}^N$ containing syzygy bundles. In a joint work with Laura Costa and
Rosa Mar{\'\i}a Mir\'o-Roig, we prove that $N$, $d$ and $n$ are as above, then
the syzygy bundle $\mathrm{Syz}(f_1,...,f_n)$ is unobstructed and it belongs to
a generically smooth irreducible component of dimension
${n\tbinom{d+N}{N}-n^2}$, if ${N\ge3}$, and
${n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2}$, if ${N=2}$.
  The results in chapter 3, for $N\ge3$, were obtained independently by Iustin
Coand\u{a} in arXiv:0909.4435.
