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.