We present new models of the magnetic fields of Uranus and Neptune, based on data provided by the Voyager II magnetic field experiment. We find the simplest models that satisfy the data, and use the observed surface heat flow as a constraint on the magnetic field structure. Our models are similar to the previously described Q3 and O8 models far from the planets, but resolve smaller-scale structure close to the planets' surfaces. The field of Neptune is much better constrained than that of Uranus, and the field in the northern hemisphere of Neptune is much better constrained than that in the southern hemisphere, of importance for studies of Neptunian northern hemisphere radio sources. Using extremal models, we show that the large dipole tilts and nondipole dominance of the fields are robust features required by the data. Scaling analysis suggests that the toroidal field that would be required for a magnetostrophic balance in the dynamo region would result in ohmic dissipation greater than the observed surface heat flow. Thus we suggest that the dynamos of Uranus and Neptune are energy limited, and that the subsequent lack of magnetostrophic balance may account for the radically different field morphologies of Uranus and Neptune compared with the Earth, Jupiter, and Saturn. ¿ American Geophysical Union 1996