  RX J0720.4-3125 is an unidentified bright soft X-ray source which shows
pulsations at a 8.39 s period and has a thermal spectrum. We present deep B and
R band images of its X-ray localization. We find one possible counterpart in
the X-ray error box, with magnitudes B=26.6+/-0.2 and R=26.9+/-0.3. The very
high X-ray to optical flux ratio confirms that this object is an isolated
neutron star. We discuss possible models and conclude that only two are
consistent with the data and at the same time are able to draw from a large
enough population to make finding one nearby likely. In our opinion the second
criterion provides a stringent constraint but appears to have been ignored so
far. The first model, suggested earlier, is that RX J0720.4-3125 is a weakly
magnetized neutron star accreting from the interstellar medium. The second is
that it is a relatively young, highly magnetized neutron star, a ``magnetar'',
which is kept hot by magnetic field decay.
