We report on the completion of the first version of a new-generation simulation code, FLASH. The FLASH code solves the fully compressible, reactive hydrodynamic equations and allows for the use of adaptive mesh refinement. It also contains state-of-the-art modules for the equations of state and thermonuclear reaction networks. The FLASH code was developed to study the problems of nuclear flashes on the surfaces of neutron stars and white dwarfs, as well as in the interior of white dwarfs. We expect, however, that the FLASH code will be useful for solving a wide variety of other problems. This first version of the code has been subjected to a large variety of test cases and is currently being used for production simulations of X-ray bursts, Rayleigh-Taylor and Richtmyer-Meshkov instabilities, and thermonuclear flame fronts. The FLASH code is portable and already runs on a wide variety of massively parallel machines, including some of the largest machines now extant.