The observations and progress in the theoretical understanding of X-ray burst sources are reviewed. Of about 150 known low mass X-ray binary systems, X-ray bursts have been observed from about 50 sources. X-ray bursts exhibit a wide variety of profiles; rise times vary from less than a second to about 10 s and decay times are in the range of 10 s to minutes with the decays much shorter at high energies than at low energies. The bursts are characterized by peak luminosities greater than 10,000 times that of the sun, peak temperatures of about 2 keV, and recur at intervals of several hours. Study of the radiation from these sources demonstrates that these sources are binary systems in which a neutron star is a member. Theoretical studies reveal that the X-ray burst results from either a combination of hydrogen and helium burning or pure helium burning in thermonuclear flashes in the envelope of an accreting neutron star.