  We study string theory propagating on R^6 times K3 by constructing
orientifolds of Type IIB string theory compactified on the T^4/Z_N orbifold
limits of the K3 surface. This generalises the Z_2 case studied previously. The
orientifold models studied may be divided into two broad categories, sometimes
related by T-duality. Models in category A require either both D5- and
D9-branes, or only D9-branes, for consistency. Models in category B require
either only D5-branes, or no D-branes at all. This latter case is an example of
a consistent purely closed unoriented string theory. The spectra of the
resulting six dimensional N=1 supergravity theories are presented. Precise
statements are made about the relation of the Z_N ALE spaces and D5-branes to
instantons in the dual heterotic string theory.
