Finite-element methods are used to study nonadhesive, frictionless contact between elastic solids with self-affine surfaces. We find that the total contact area rises linearly with the load at small loads. The mean pressure in the contact regions is independent of load and proportional to the root-mean-square slope of the surface. The constant of proportionality is nearly independent of the Poisson ratio and roughness exponent and lies between previous analytic predictions. The contact morphology is also analyzed. Connected contact regions have a fractal area and perimeter. The probability of finding a cluster of area a(c) drops as a(c)(-tau) where tau increases with a decrease in roughness exponent. The distribution of pressures shows an exponential tail that is also found in many jammed systems. These results are contrasted to simpler models and experiments.