The Center for Astrophysical Thermonuclear Flashes at the University of Chicago — a collaboration between scientists located primarily at Chicago and at Argonne National Laboratory — studies the long-standing problem of thermonuclear flashes on the surfaces of compact stars such as neutron stars and white dwarf stars, and in the interior of white dwarf stars. A key objective of the Center is to build a new-generation astrophysical simulation code for computing problems relating to X-ray bursts, novae, and Type Ia supernovae. The capabilities of the code are extensive; the code can treat problems in astrophysics ranging from star formation in molecular clouds to large-scale structure formation in cosmology. In this talk I will highlight a number of recent scientific results obtained by researchers at the Center using the Flash code and new modules that are being developed for the code. These include: (1) 1-D relativistic calculations of accretion onto neutron stars; (2) 2-D semi-relativistic MHD calculations of accretion onto magnetic white dwarfs; (3) 2-D and 3-D simulations of wind-wave mixing in the envelope of pre-novae white dwarfs, which have led to a new physical picture of nova outbursts; (4) flame modeling within the context of Type Ia supernovae; and (5) the first 3-D simulations of the deflagration phase of Type Ia supernovae for an entire white dwarf star.