  Observations of cooling neutron stars allow to measure photospheric radii and
to constrain the equation of state of nuclear matter at high densities. In this
paper we concentrate on neutron stars, which show thermal (photospheric) X-ray
emission and have measured distances. After a short summary of the radio
pulsars falling into this category we review the observational data of the 7
radio quiet isolated neutron stars discovered by ROSAT which have been studied
in detail by Chandra, XMM-Newton and optical observations. Their spectra show
blackbody temperatures between 0.5 and 1 million Kelvin and an optical excess
of a factor of 5-10 over the extrapolation of the X-ray spectrum. Four of these
sources show periodicities between 3.45 and 11.37 s, indicating slow rotation.
The pulsed fractions are small, between 6 and 18 %. The magnetic fields derived
from spin down and/or possible proton cyclotron lines are of the order
10$^{13}-10^{14}$ G. We then discuss RX J1856.5-3754 in detail and suggest that
the remarkable absence of any line features in its X-ray spectrum is due to
effects of strong magnetic fields ($\sim 10^{13}$ G). Assuming blackbody
emission to fit the optical and X-ray spectrum we derive a conservative lower
limit of the ``apparent'' neutron star radius of 16.5 km $\times$ (d/117 pc).
This corresponds to the radius for the ``true'' radius of 14 km for a 1.4
M$_{\odot}$ neutron star, indicating a stiff equation of state at high
densities. A comparison of the result with mass-radius relations shows that in
this case a quark star or a neutron star with a quark matter core can be ruled
out with high confidence.
