Type I cosmic X-ray bursts are widely thought to result from thermonuclear flashes in the surface layers of accreting neutron stars. The thermonuclear-flash model is able to account for a wide variety of observed burst phenomena. However, a number of theoretical and observational problems persist. Foremost among these are related to the existence of apparent luminosities in excess of the Eddington limit in many observed bursts. Under circumstances of very high luminosity, the theoretical treatment of radiative transfer and radiatively driven mass loss at the neutron-star surface becomes complex. We discuss the current status of theoretical work in this area and the prospects for developing an understanding of the remaining phenomenological problems: (a) What is the nature of the precursor in some fast transients and of the twin-peak structure observed in some shorter bursts? (b) Do some bursts actually achieve super-Eddington luminosities, and if so, how? (c) What are the characteristics of quasi-static mass loss in the presence of such luminosities? (d) How is the color temperature related to the effective temperature? and, (e) What are the implications concerning the Galactic distance scale?