Abstract We study string theory propagating on R 6 × K3 by constructing orientfolds of Type IIB string theory compactified on the T 4/ Z N orbifold limits of the K3 surface. Thi 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.