Novae are cataclysmic variable binary systems in which a white dwarf (WD) primary is accreting material from a low mass companion. The importance of this accretion takes on added significance if the WD can increase its mass to reach the Chandrasekhar limit thus exploding as a Type Ia supernova. In this study we accrete material of Solar composition onto carbon/oxygen (CO) WDs of 0.70, 1.00 and 1.35 M⊙ with accretion rates from 1.6×10-10 to 1.6×10-6 M⊙ yr-1. We have utilized the MESA stellar evolution code for our modeling and evolve them for many nova cycles or, in some cases, evolution to a red giant stage. Differing behaviors occur as a function of both the WD mass and the accretion rate. For the lower WD masses, the models undergo recurrent hydrogen flashes at low accretion rates; for higher accretion rates, steady-burning of hydrogen occurs and eventually gives way to recurrent hydrogen flashes. At the highest accretion rates, these models go through a steady-burning phase but eventually transition into red giants. For the highest WD mass recurrent hydrogen flashes occur at lower accretion rates but for higher rates the models exhibit steady-burning interspersed with helium flashes. We find that for all our models that undergo recurrent hydrogen flashes, as well as the steady-burning models that exhibit helium flashes, the mass of the WD continues to grow toward the Chandrasekhar limit. These results suggest that the accretion of Solar abundance material onto CO WDs in cataclysmic variable systems, the single degenerate scenario, is a viable channel for progenitors of Type Ia supernova explosions.