We have performed Monte Carlo (MC) simulations on systems of L×L classical planar unit spins on square lattices, for L=6, 15, 30, 60, 90, and 200. The interaction between any two given spins S1 and S2 is given by -JS1 S2 if S1 and S2 are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (Tc). We have found that the diffusion constant for vortex motion is given at Tc by D0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically 105 MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of Tc=0.89J/kB, the value of =0.5 0.1, and the value of c=0.24 0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory. © 1986 The American Physical Society.