HELIUM-RICH ACCRETION ONTO WHITE DWARFS (WDS) OCCURS IN A VARIETY OF BINARY SYSTEMS AND GIVES RISE TO A RANGE OF THERMONUCLEAR PHENOMENA INCLUDING CONVECTIVELY-BURNING HELIUM NOVAE SIMILAR TO HYDROGEN-POWERED CLASSICAL NOVAE AND SUBSONIC DEFLAGRATIONS THAT YIELD FAINT RAPIDLY-EVOLVING ".IA" SUPERNOVAE. ALTHOUGH PROGENITOR BINARIES AND CANDIDATE OUTBURSTS FOR BOTH OF THESE TYPES OF TRANSIENTS HAVE BEEN OBSERVED NEITHER CLASS HAS BEEN STUDIED IN MUCH THEORETICAL DETAIL. THE MOST LUMINOUS POTENTIAL HELIUM-BURNING OUTCOMES ARE TYPE IA SUPERNOVAE (SNE IA) WHICH PLAY AN ESSENTIAL ROLE IN DETERMINING THE ACCELERATING EXPANSION OF THE UNIVERSE PRODUCE HALF OF THE UNIVERSE'S HEAVY METALS AND SERVE AS TESTBEDS FOR STELLARHYDRODYNAMICS NUCLEAR PHYSICS BINARY STELLAR EVOLUTION AND MANY OTHER FIELDS OF ASTROPHYSICS. HOWEVER DESPITE THEIR IMPORTANCE AND SEVERAL DECADES OF THEORETICAL INVESTIGATION THE NATURE OF THE STELLAR SYSTEMS THAT GIVE RISE TO THESE EXPLOSIONS REMAINS A MYSTERY. AN EXPLOSIVE HELIUM DETONATION ON THE SURFACE OF A WD THAT IN TURN YIELDS A CARBON-BURNING DETONATION IN THE CORE IS PERHAPS THE MOST PROMISING SN IA PROGENITOR CANDIDATE BUT IS ALSO ONE OF THE LEAST EXPLORED DUE TO ITS RELATIVELY RECENT EMERGENCE. OUR PROPOSED WORK COMPRISES THE FIRST DETAILED STUDIES OF ALL THREE OF THESE HELIUM-BURNING OUTCOMES: CONVECTIVELY-EFFICIENT HELIUM NOVAE SUBSONIC DEFLAGRATIONS AND SUPERSONIC DETONATIONS. WE WILL USE PREVIOUSLY DEVELOPED TECHNIQUES AND SOFTWARE FOR SIMULATIONS OF PROGENITOR EVOLUTION DEFLAGRATION AND DETONATION HYDRODYNAMICS NUCLEAR REACTION KINETICS AND RADIATIVE TRANSFER. THESE WILL INCLUDE THE FIRST HYDRODYNAMIC STUDIES OF HELIUM NOVAE AND THE FIRST MULTI-DIMENSIONAL LARGE NUCLEAR REACTION NETWORK SIMULATIONS OF HELIUM DEFLAGRATIONS AND DETONATIONS INCLUDING THE EFFECTS OF COMPOSITION POLLUTION ON THE FLAME PROPAGATION. DATA PRODUCED AND SOFTWARE DEVELOPED IN THIS PROJECT WILL BE OPEN ALLOWING FOLLOW-UP STUDIES BY OTHER GROUPS AS WELL AS SUPPORTING INDIRECTLY RELATED WORK THAT USES TOOLS WE WILL DEVELOP IN THE COURSE OF OUR PROPOSED EFFORT. OUR PROPOSED RESEARCH REPRESENTS THE FIRST DETAILED EXPLORATIONS OF THE DIVERSE OUTCOMES OF HELIUM BURNING WHICH HAVE THE POTENTIALTO YIELD NEW AND RELATIVELY UNSTUDIED CLASSES OF TRANSIENTS AND TO EXPLAIN ONE OF THE MOST IMPORTANT AND ENDURING MYSTERIES OF ASTROPHYSICS THE PROGENITORS OF SNE IA. BECAUSE OF THE BROAD ASTROPHYSICAL SIGNIFICANCE OF SNE IA AND OTHER EXPLOSIVE TRANSIENTS OUR RESEARCH INTO THEIR PROGENITORS AND OUTBURSTS WILL HAVE DIRECT BEARING ON CURRENT FUTURE AND POTENTIAL NASA MISSIONS SUCH AS HST CHANDRA NUSTAR SWIFT JWST AND WFIRST-AFTA.