As many as $10^9$ neutron stars populate the Galaxy, but only $\approx 10^3$
are directly observed as pulsars or as accreting sources in X-ray binaries. In
principle also the accretion of the interstellar medium may make isolated
neutron stars shine, and their weak luminosity could be detected in soft
X-rays. Recent ROSAT observations have convincingly shown that neutron stars
accreting from the interstellar medium are extremely rare, if observed at all,
in contrast with earlier theoretical predictions. Until now two possible
explanations for their elusiveness have been proposed: their velocity
distribution may peak at $\sim 200-400 {\rm km s}^{-1}$, as inferred from
pulsar statistics, and this would severely choke accretion; the magnetic field
may decay on timescales $\sim 10^8-10^9$ yr, preventing a large fraction of
neutron stars from entering the accretor stage. The search for accreting
neutron stars has produced up to now a handful of promising candidates. While
little doubt is left that these objects are indeed isolated neutron stars, the
nature of their emission is still controversial. In particular accreting
objects can be confused with much younger, cooling neutron stars. However, a
combination of observations and theoretical modeling may help in discriminating
between the two classes.Comment: 22 pages Latex, 3 Figures, Invited Review to be published to PAS