  We study the properties of spin systems realized by cold polar molecules
interacting via dipole-dipole interactions in two dimensions. Using a spin wave
theory, that allows for the full treatment of the characteristic long-distance
tail of the dipolar interaction, we find several anomalous features in the
ground state correlations and the spin wave excitation spectrum, which are
absent in their counterparts with short range interaction. The most striking
consequence is the existence of true long-range order at finite temperature for
a two-dimensional phase with a broken U(1) symmetry.
