Using Monte Carlo simulations we have found a significant logarithmic size dependence for the density of defects, the shear elastic constant, and the long-range angular order in two-dimensional solids close to their melting points. These size dependencies were present for both hard discs and the repulsive inverse 12th-power potential. Our data suggest that these size dependencies may also be present at higher densities further from the melting density. Simulations very close to melting suggest that the melting densities in the literature are incorrect. In this paper we present the results from a series of computer simulations on two-dimensional solids. The aim of these simulations was to determine the detailed physical properties of these systems near their melting points. For the purposes of this work we accepted that the melting transition was weakly first order, and we also accepted the best previous estimates of the melting points. Our work was confined to potentials. We have generated very detailed data on the hard-disc (HD) system and the repulsive-inverse 12th-power (R-12) systems. ' We will show that it is possible to map these systems onto each other, thus our results have some degree of universality for potentials. Our basic aim was to examine the behavior of these systems near melting with the goal of shedding some light on the complexities of simulations in this region of the thermodynamic space of the systems. Since the results of direct simulations of the melting-freezing transition tend to point toward a weak first-order transition, ' we thought it worthwhile to carry out a fairly exhaustive examination of the solid phase to see whether our data could say anything, even indirectly, about the likely nature of the transition. We believe that a definitive statement about the nature of the transition, based on computer simulations, must await the development of a version of the Monte Carlo (MC) renormalization group method which is applicable to particle systems. We can easily summarize our results. There are strong size dependences and novel phenomena near melting in the systems we have studied. Our data cast considerable doubt on the validity of simulations of the transition which do not include at least 16000 particles. Our system sizes ranged from 1024 to 16384 particles, and we were able to carry out MC runs of up to two million sweeps for these systems. 'We found that both the large system size and the long runs are essential if the full range of phenomena were to be revealed. As an essential first step, we determined the pressure of the systems very near melting. Except at one density, all our pressure measurements showed size-independent results — the systems appeared to be in the thermodynamic limit. At one density, about 0.5% from melting, we found anomalous size dependencies for the HD system. These results may be due to a mislocation of the melting density. They may also have a more interesting interpretation. However, other properties such as the density of topological defects, the shear elastic constant, the particle displacements, and long-range angular order showed strong size dependencies and anomalous behavior 1% to 2% from melting. Some of these results have already been observed by Toxvaerd. In most instances we have been able to make a more thorough exploration of the configuration space of these systems, and enlarge the data by simulating larger systems. Our simulations are conducted at constant temperature and density using the standard Metropolis MC algorithm. We used periodic boundary conditions. A word of caution with respect to the simulation of the HD system is necessary. For this system, MC moves are accepted or rejected depending on whether two discs do or do not overlap. Since there are no moves in which the energy increases, no additional random number is generated to determine acceptance or rejection. This essential difference between the HD and other stiff potentials means that a very different and predictable sequence of pseudorandom numbers is used in the HD system. Initially, we overlooked the fact and were led astray by long period correlations in our pseudorandom number sequences. We were able to overcome the difficulty in two distinct ways which are described in the Appendix. For the HD system we were able to make runs with two million sweeps. The code for the R-12 system was considerably slower. For this system we made runs of 300000 sweeps. For both systems the codes were very carefully optimized for the FPS M64 processors which are part of the Cornell National Supercomputer Facility. For both systems we computed the pressure, density of topological defects, the mean-square displacements of the particles, and the local and long-range angular order. For the R-12 system we also computed the shear elastic constant.