In this paper we consider two contact problems for an anisotropic elastic half-space in which the stress is independent of one of the Cartesian co-ordinates. The results hold for the most general anisotropy in which no symmetry elements of the material are assumed. Problems of this type have been considered by Brilla [1], Clements [2], Galin [3], Green and Zerna [4] and Milne-Thomson [5], but the work of these authors is only applicable to a restricted class of anisotropic materials. We begin in section 1 by deriving some fundamental equations for the stress and displacement. It will be noted that at an early stage (equation (7)) the roots of a sextic polynomial are required. Since these cannot be obtained explicitly any application of the general theory must, of necessity, be numerical. However the calculation of the stress and displacement is simplified if certain symmetry elements of the material are assumed and the way in which these simplifications occur is indicated in section 2. In sections 3 and 4 we consider contact problems in which the elastic half-space is indented by a rigid body. For the problem considered in section 3, the rigid body is assumed to be able to move relative to the surface of the half-space, while for the problem considered in section 4 it is assumed to be linked to the half-space.