I review our theoretical understanding of thermonuclear flashes on accreting neutron stars, concentrating on comparisons to observations. Sequences of regular Type I X-ray bursts from GS 1826-24 and 4U 1820-30 are very well described by the theory. I discuss recent work which attempts to use the observed burst properties in these sources to constrain the composition of the accreted material. For GS 1826-24, variations in the burst energetics with accretion rate indicate that the accreted material has solar metallicity; for 4U 1820-30, future observations should constrain the hydrogen fraction, testing evolutionary models. I briefly discuss the global bursting behavior of burst sources, which continues to be a major puzzle. Finally, I turn to superbursts, which naturally fit into the picture as unstable carbon ignition in a thick layer of heavy elements. I present new time-dependent models of the cooling tails of superbursts, and discuss the various interactions between superbursts and normal Type I bursts, and what can be learned from them.

11 pages, 6 figures; to appear in Proc. of the 2nd BeppoSAX Meeting: "The Restless High-Energy Universe" (Amsterdam, May 5-8, 2003), E.P.J. van den Heuvel, J.J.M. in 't Zand, and R.A.M.J. Wijers (Eds)