  We prove the irreducibility of the moduli space of rank 2 semistable torsion
free sheaves (with a generic polarization and any value of c_2) on a K3 or a
del Pezzo surface. In the case of a K3 surface, we need to prove a result on
the connectivity of the Brill-Noether locus for singular curves on the surface.
In the case of a del Pezzo surface, we reduce the problem to the case of P^2 by
first relating the moduli spaces of the plane and the blown-up plane, and then
studying how the moduli space changes when we change the polarization.
