  Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we
compute the cohomology of a general sheaf in the moduli space of Gieseker
semistable sheaves. We also classify the Chern characters for which the general
sheaf in the moduli space is non-special, i.e. has at most one nonzero
cohomology group. Our results hold for arbitrary polarizations, slope
semistability, and semi-exceptional moduli spaces. When the degree of $X$ is at
least 3, we further show our construction of certain vector bundles implies the
existence of stable and semistable sheaves with respect to the anti-canonical
polarization.
