This paper concerns hypercomplex manifolds ([3, § 6],[4, pp. 137-139]) and quaternionic manifolds ([3, § 1],[4, pp. 135-136]), which are manifolds with a GL (n, H)-and a GL (n, H) H [*-structure respectively, preserved by a torsion-free connection. It is in two parts, and each part presents a way of constructing compact examples of these manifolds. In the first part a method is given similar to those used by Gibbons and Hawking to construct hyper-Kahler manifolds and by LeBrun [2] to construct scalar-flat Kahler surfaces. It will be shown that given a hypercomplex or quaternionic manifold M, a Lie group G, an action Ψ of G on M that preserves the structure, and a Ψ-invariant quaternionic G-connection on a principal G-bundle P over M, one can, subject to a certain condition, define a new hypercomplex or quaternionic manifold N that is M" twisted by" the G-bundle P. Here a quaternionic connection is one satisfying a curvature condition that naturally generalizes the instanton equations in the four-dimensional case. In the second part the theory of homogeneous hypercomplex and quaternionic manifolds will be described. This is based upon the theory of homogeneous complex manifolds given in [5],[8]. The case of homogeneous hypercomplex structures on groups has already been described by Spindel et al.[6].

Both of these methods give many compact, nonsingular, simplyconnected hypercomplex and quaternionic manifolds in dimensions greater than four, which are not products or joins of other manifolds, and are not (even locally) hyper-Kahler or quaternionic Kahler.(That is, the structure group cannot be reduced to SP («) or SP («) SP (1).) We believe that these are the first such examples to be described, other than the homogeneous hypercomplex groups in [6].