The advent of X-ray telescopes (CHANDRA, XMM-Newton) in recent years has made the detection and detailed study of the surface emission from isolated neutron stars (including radio pulsars, radio-quiet neutron stars and magnetars) a reality. I will discuss current observational status and related theoretical issues. Possible topics include: (1) Effect of strong-field quantum electrodynamics on radiative transfer and how recent observations of magnetars may have already "proved" that the QED effect is at work; (2) The origin of line features (e.g. in 1E1207) and what we can learn from them; (3) Possible surface condensation in strong magnetic fields and the origin of featureless spectra observed in many neutron stars; (4) Nucleosynthesis in nascent neutron stars and how it may shape the star's surface composition; (5) Polarization signatures of neutron star X-ray emission, and the prospect of using X-ray polarimetry (under development) to test fundamental physics in strong magnetic fields and probe the nature of magnetars.