Thermonuclear X-ray bursts from the surface of accreting neutron stars are the most common astrophysical explosions in our galaxy. They provide a unique window into the physics of neutron stars, the physics of matter under extreme conditions, and the physics of astrophysical thermonuclear explosions. X-ray bursts are powered by a broad range of nuclear reactions that need to be understood to interpret observations. The relevant nuclei are mostly neutron deficient and unstable, and thus experimental information and theoretical understanding is limited and an active area of research in nuclear science. We review the current status of the nuclear physics of X-ray bursts, with special emphasis on new experimental and theoretical information on a large number of reaction rates. As such we provide an overview of the broad experimental and theoretical methods currently used to advance the nuclear physics of X-ray bursts. The new information is used to update the public JINA REACLIB database with 32 new reaction rates based on experimental information, and a new dataset of theoretical statistical model reaction rates where no experimental information is available. Using several models for X-ray bursts that are powered by mixed hydrogen and helium burning, we take advantage of the updated nuclear data to review the current understanding of the nuclear reaction sequences in such X-ray bursts, the modeling of light curves, and predictions of the composition of nuclear ashes.

92 pages, 21 figures, accepted at Physics Reports