  Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an
ample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli space
of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes
$c_i(E)=c_i$ for $i=1, ..., s:=min\{r,n\}$. We define the Brill-Noether
filtration on $M_{X,H}(r;c_1, ..., c_{s})$ as $W_{H}^{k}(r;c_1,..., c_{s})= \{E
\in M_{X,H}(r;c_1, ..., c_{s}) | h^0(X,E) \geq k \}$ and we realize
$W_{H}^{k}(r;c_1,..., c_{s})$ as the $k$th determinantal variety of a morphism
of vector bundles on $M_{X,H}(r;c_1, ..., c_{s})$, provided $H^i(E)=0$ for $i
\geq 2$ and $E \in M_{X,H}(r;c_1, ..., c_{s})$. We also compute the expected
dimension of $W_{H}^{k}(r;c_1,..., c_{s})$. Very surprisingly we will see that
the Brill-Noether stratification allow us to compare moduli spaces of vector
bundles on Hirzebruch surfaces stables with respect to different polarizations.
We will also study the Brill-Noether loci of the moduli space of instanton
bundles and we will see that they have the expected dimension.
