Rough surfaces are modeled as two-dimensional, isotropic, Gaussian random processes, and analyzed with the techniques of random process theory. Such surface statistics as the distribution of summit heights, the density of summits, the mean surface gradient, and the mean curvature of summits are related to the power spectral density of a profile of the surface. A detailed comparison is made of the statistics of the surface and those of the profile, and serious differences are found in the distributions of heights of maxima and in the mean gradients. Techniques for analyzing profiles of random surfaces to obtain the parameters necessary for the analysis of the surface are discussed. Extensions of the theory to nonisotropic Gaussian surfaces are indicated.