We present timing measurements, astrometry, and high-resolution spectra of a number of nearby, thermally emitting, isolated neutron stars. We use these to infer magnetic field strengths and distances, but also encounter a number of puzzles. We discuss three specific ones in detail: (i) For RX J0720.4−3125 and RX J1308.6+2127, the characteristic ages are in excess of 1 Myr, while their temperatures and kinematic ages indicate that they are much younger; (ii) For RX J1856.5−3754, the brightness temperature for the optical emission is in excess of that measured at X-ray wavelengths for reasonable neutron-star radii; (iii) For RX J0720.4−3125, the spectrum changed from an initially featureless state to one with an absorption feature, yet there was only a relatively small change in T eff . Furthermore, we attempt to see whether the spectra of all seven sourced, in six of which absorption features have now been found, can be understood in the context of strongly magnetised hydrogen atmospheres. We find that the energies of the absorption features can be reproduced, but that the featureless spectra of some sources, especially the Wien-like high-energy tails, remain puzzling.