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Thermonuclear burning in the matter accumulated§on the surface of an accreting neutron star is§unstable under certain conditions, which leads to§powerful explosions also called Type I X-ray§bursts. This book covers the theory of§spectral formation in the atmospheres of neutron§stars during these violent events as well as its§application to observational data on X-ray bursts. §The main goal of the presented study is a proper§measurement of the fundamental properties for§neutron stars. We take both§analytical and numerical approaches to the problem of§radiative transfer in X-ray illuminated atmospheres§which account for all crucial physical factors such§as Comptonization, free-free absorption and emission.§We apply our models to the data collected with NASA's§Rossi X-ray Timing Explorer mission in order to§constrain neutron star masses and radii for five§X-ray burster 4U 1728-34, 4U 1820-30, Cygnus X-2, EXO§0748-676 and GS 1826-24. The developed neutron star§mass-radius measurement method has a great potential§for application to new and archival X-ray data. It§can provide scientists with a vital information on§various aspects of physics of compact objects.