The physical contact between two rough surfaces is referred to as a ''joint,'' and the deformation of such a joint under normal stress is called the ''joint closure.'' Toward better understanding of joint closure, we present a theory of contact between two random nominally flat, elastic surfaces. This theory is a more general form of a theory presented previously by others for the elastic contact of a rough surface and a flat surface. In agreement with the previous theory we show that the joint closure property depends as much on the details of the surface topography as on the elastic properties of the material. To apply these results by using linear surface profiles requires mapping of profile information to three dimensions. The mapping techniques described here require the probability density function for the contacting surfaces to be approximated well by either a Gaussian distribution or an inverted chi-square distribution. Laboratory experiments on ground surfaces of glass samples were done to test the theory. Both joint closure and surface topography were measured. In most cases, experimental results agreed quantitatively with predictions of the theory. However, in experiments on the smoothest surfaces, sample preparation problems often resulted in surfaces with a domed shape. These surfaces did not fit the assumptions of the theory, but the observed deviation from the theory were consistent with this domed shape. Surface topography measurements suggest that many surfaces are statistically similar. This implies that the success of the theory in predicting joint closure does not depend on a particular sample preparation technique. Therefore the theory in predicting joint closure does not depend on a particular sample preparation technique. Therefore the theory should be valid for all nominally flat elastic surfaces. The form of the power spectrum implies that the surface topography and thus the joint closure depend on sample size.