A convergent low-temperature expansion for a variety of models of two-dimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity in z = e^(-B). In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the s01id-on-solid and discrete Gaussian models.