  We present results of large-scale Monte Carlo simulations of the 2D classical
x-y model on the square lattice. We obtain high accuracy results for the
superfluid fraction and for the specific heat as a function of temperature, for
systems of linear size L up to 4096. Our estimate for the superfluid transition
temperature is consistent with those furnished in all previous studies. The
specific heat displays a well-defined peak, whose shape and position are
independent of the size of the lattice for L > 256, within the statistical
uncertainties of our calculations. The implications of these results on the
interpretation of experiments on adsorbed thin films of He-4 are discussed.
