Type I X-ray bursts are thermonuclear explosions that occur in the envelopes of accreting neutron stars. Detailed observations of these phenomena have prompted numerous studies in theoretical astrophysics and experimental nuclear physics since their discovery over 35 years ago. In this review, we begin by discussing key observational features of these phenomena that may be sensitive to the particular patterns of nucleosynthesis from the associated thermonuclear burning. We then summarize efforts to model type I X-ray bursts, with emphasis on determining the nuclear physics processes involved throughout these bursts. We discuss and evaluate limitations in the models, particularly with regard to key uncertainties in the nuclear physics input. Finally, we examine recent, relevant experimental measurements and outline future prospects to improve our understanding of these unique environments from observational, theoretical and experimental perspectives. (A. Parikh).1 See http://www.sron.nl/∼jeanz/bursterlist.html for a list of known Galactic bursting sources. 2 Most X-ray sources are named using letters from the satellites that discovered them (e.g., 4U stands for the fourth catalogue of the satellite Uhuru, the first X-ray observatory) and numbers corresponding to their coordinates in right ascension (1820 stands for 18h20min) and declination (-30 deg) in the sky. These sources may also be named after the constellation where they are located and the order of discovery: 4U 1820-30 is also known as Sgr X-4, the fourth X-ray source discovered in the constellation Sagittarius. To complicate matters further, even within one of these naming conventions a single source may have two names. For example, the source 4U 1820-30 has been referred to in the literature as 3U 1820-30 as well.3 1 Crab = 2.4×10 −8 erg s −1 cm −2 in the 2-10 keV band.2 to 2003 [9]), and the Rossi X-ray Timing Explorer (RXTE, operating from 1995(RXTE, operating from to 2012). RXTE, with its large collecting area and capability for high temporal resolution X-ray analysis, has observed more than 1000 bursts from about 50 sources [11]. Soon after the discovery of X-ray bursting sources, it was proposed that the explosions could be powered by thermonuclear runaways on the surface of accreting neutron stars [12,13] (see Section 3.1). The discovery of the Rapid Burster in 1976 [14], with burst recurrence times as short as 10 s, complicated matters. The quick succession of flashes observed from this source seemed incompatible with the proposed mechanism, and moreover, did not match the general pattern shown by the other bursting sources. A classification of type I and type II bursts was therefore established [15], the former associated with thermonuclear flashes and the latter linked to accretion instabilities. The Rapid Burster is one of only two known sources showing type II bursts, together with the Bursting Pulsar (GRO 1744-28); furthermore, the Rapid Burster is the only source giving both type I and type II X-ray bursts [16]. Type I bursts display a sp...