The possibility for direct investigation of thermal emission from isolated neutron stars was opened about 25 years ago with the launch of the first X-ray observatory, Einstein. A significant contribution to this study was provided by ROSAT in 1990s. The outstanding capabilities of the currently operating observatories, Chandra and XMM-Newton, have greatly increased the potential to observe and analyze thermal radiation from the neutron star surfaces. Confronting observational data with theoretical models of thermal emission, presumably formed in neutron star atmospheres, allows one to infer the surface temperatures, magnetic fields, chemical composition, and neutron star masses and radii. This information, supplemented with model equations of state and neutron star cooling models, provides an opportunity to understand the fundamental properties of the superdense matter in the neutron star interiors. I review the current status and most important results obtained from modeling neutron star thermal emission and present selected Chandra and XMM-Newton results on thermal radiation from various types of these objects: ordinary radio pulsars with ages ranging from about 2 kyr to 20 Myr (J1119-6127, Vela, B1706-44, J0538+2817, B2334+61, B0656+14, B1055-52, Geminga, B0950+08, J2043+2740), millisecond pulsars (J0030+0451, J2124-3358, J1024-0719, J0437-4715), putative pulsars (CXOU J061705.3+222127, RX J0007.0+7302), central compact objects in supernova remnats (in particular, 1E 1207.4-5209), and isolated radio-quiet neutron stars.
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