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Classical nova explosions and type I X-ray bursts are the most frequent types of thermonuclear stellar explosions in the Galaxy. Both phenomena arise from thermonuclear ignition in the envelopes of accreting compact objects in close binary star systems. Detailed observations of these events have stimulated numerous studies in theoretical astrophysics and experimental nuclear physics. We discuss observational features of these phenomena and theoretical efforts to better understand the energy production and nucleosynthesis in these explosions. We also examine and summarize studies directed at identifying nuclear physics quantities with uncertainties that significantly affect model predictions.