The nature of the melting transition in two dimensions is critically dependent on the core energy of dislocations. In this paper, we report calculations of the core free energy and the core size of dislocations in two-dimensional solids of systems interacting via square well, hard disk, and r-12 potentials. In all cases, we find that the dislocation core free energy is such that, at the densities studied, the density of free dislocation density is extremely low. We find that the core energies and core sizes are considerably smaller for the r-12 system than for the other systems studied. This illustrates the fact that, for hard-core systems, elastic continuum theory breaks down, even for relatively small strains.