Neutron stars in mass-transferring binaries are accreting the hydrogen and helium rich matter from the surfaces of their companions. This article simply explains the physics associated with how that material eventually fuses to form heavier nuclei and the observations of the time dependent phenomena (such as Type I X-ray bursts) associated with the thermally unstable thermonuclear reactions. We explain how the outcome depends on the composition of the accreting matter, the accretion rate and the mass, radius and thermal state of the neutron star. We also introduce many new analytic relations that are convenient for comparisons to both observations and computational results. After explaining nuclear burning for spherically symmetric accretion onto neutron stars, we discuss the possibility of asymmetric burning. In particular, we discuss some of the mysteries from EXOSAT observations of Type I X-Ray bursts and how the solution to these puzzles may lie in considering the lateral propagation of nuclear burning fronts around the star. Fully understanding this problem requires knowledge of parameters previously neglected such as the distribution of fresh fuel on the star, the magnetic field strength, and the stellar rotation. Recent RXTE observations of bursters may finally tell us some of these parameters.