We have begun new studies of the evolution of thermonuclear runaways (TNRs) in the accreted envelopes of white dwarfs (WDs). Here we focus on the recent outbursts of RS Oph (2006), U Sco (2010) and T Pyx (2011). U Sco explodes about every 10 years and the ejected material from the WD is helium rich. It has a short orbital period for recurrent novae (RNe) but the secondary is likely to be evolved. The WD is thought to be close in mass to the Chandrasekhar limit. T Pyx has just suffered its first outburst since 1966 and it was predicted to never experience another outburst. It has a short orbital period and has formed dust in the ejecta as this paper was being written. One important question is the secular evolution of the WD. Do the repeated outbursts cause the WD to gain or lose mass? If it is gaining mass, it could eventually reach the Chandrasekhar limit and become a Type Ia supernova (SNe Ia) if it can hide the hydrogen and helium in the system. Here, we report on our latest studies of TNRs in accreted envelopes on WDs using a variety of initial WD masses, luminosities, and mass accretion rates. Of great importance to our conclusions, we assume a solar composition (Lodders abundance distribution). We use our 1-D hydro code, NOVA, that includes the Hix and Thielemann nuclear reaction network, the Iliadis reaction rate library, the Timmes equation of state, OPAL opacities, and the new convection of Arnett, Meakin, and Young. We report on the amount of ejected mass, evolution time to explode, and whether or not the WD is growing or losing mass.