  Strange duality is shown to hold over generic $K3$ surfaces in a large number
of cases. The isomorphism for elliptic $K3$ surfaces is established first via
Fourier-Mukai techniques. Applications to Brill-Noether theory for sheaves on
$K3$s are also obtained. The appendix written by Kota Yoshioka discusses the
behavior of the moduli spaces under change of polarization, as needed in the
argument.
