Neutron stars are one of the most dense objects in the Universe. However, the exact description of the equation of state (EoS) of the cold ultra-dense matter inside them is still a mystery. In this thesis, we measure the size of some neutron stars using astrophysical observations of X-ray bursts that are produced by thermonuclear runaways in the uppermost layers of the star. By measuring the size, we can then set constraints on the nuclear physics of the interiors and ultimately on the EoS of the cold dense matter. 

The size measurements are done by comparing the cooling of the neutron star surfaces after the bursts to theoretical atmosphere model calculations. Hence, accurate modeling of the emergent radiation from the atmospheres is needed. In the first part of this thesis, I have studied how the emergent spectra differ if the atmosphere is enriched with nuclear burning ashes from the bursts. This gives us new tools to understand and interpret the X-ray burst observations. In addition, I have shown how the emerging radiation is modified when it originates from rapidly rotating oblate neutron stars. 

Furthermore, we must also be careful in selecting only those bursts that are not influenced by the infalling material. In the second part of the thesis, I have focused on studying the astrophysical environments of the X-ray bursts in order to quantify the effect of accretion on the mass and radius measurements. Importantly, it is shown that only the bursts that occur during the low-accretion-rate (hard) state can be used for the size determination because otherwise the accretion flow might influence the cooling of the stellar surface. 

After taking these steps into account, it is possible to set constraints on the mass, radius, distance, and atmosphere composition of neutron stars exhibiting X-ray bursts. In the third part of the thesis, I have used the aforementioned models and methods to constrain the mass and radius of neutron stars using the hard state X-ray bursts. The method has been applied to three neutrons stars in low-mass X-ray binary systems 4U 1702-429, 4U 1724-307, and SAX J1810.8-260 for which the radius is measured to be between 10.9 - 12.4 km (68% credibility). The newly computed atmosphere models have also been used to detect a presence of burning ashes in the atmosphere of the neutron star in HETE J1900.1-2455. Later on, an improved Bayesian method of fitting the atmosphere models directly to the observed spectra has also improved the radius constraints of 4U 1702-429 to R = 12.4 +- 0.4 km (68% credibility). These results are in a good agreement with the current nuclear physical predictions and demonstrate how astrophysical measurements can be used to gauge the unknown nuclear physics of neutron stars.