It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle E_{d_1,...,d_n} on P^N defined as the kernel of a general epimorphism

\phi: O(-d_1)\oplus\cdots\oplus O(-d_n) -> O

is (semi)stable. In this talk, attention is restricted to the case of syzygy bundles Syz(f_1,...,f_n) on P^N associated to n generic forms f_1,...,f_n\in K[X_0,...,X_N] of the same degree d, for N\ge 2. The first goal is to prove that 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 P^N containing syzygy bundles. In a joint work with Laura Costa and Rosa María Miró-Roig, we prove that if N\ne 3 and d and n are as above, then the syzygy bundle 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\ge 4, and n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2}, if N=2.  Note: the case N\ge 3 for stability of syzygy bundles was independently solved by Iustin Coand\v{a} [arXiv:0909.4455].