A theory of contact between two elastic rough surfaces is presented that predicts, from knowledge of the surface topography, both the normal and shear deformation. The theory was modified from one developed by others in order to properly account for initial overlap of the two surfaces and to allow for calculation of shear stiffnesses under no-slip conditions. Since the theory is critically dependent on the surface topography, a new method is introduced for the derivation of the real topographic parameters. The result of theoretical calculation shows that the normal closure nonlinearly increases with normal stress and is larger for rougher surfaces and that the shear stiffness also nonlinearly increases with normal stress and is greater for smoother surfaces than rougher ones. The verification of the theory will be made by comparing it with experimental data in a companion paper. ¿ American Geophysical Union 1989