It has been argued that low-luminosity dwarf galaxies are the dominant source of ionizing radiation during cosmological reionization . The fraction of ionizing radiation that escapes into the intergalactic medium from dwarf galaxies with masses less than \sim 10 ^ { 9.5 } solar masses plays a critical role during this epoch . Using an extensive suite of very high resolution ( 0.1 pc ) , adaptive mesh refinement , radiation hydrodynamical simulations of idealized and cosmological dwarf galaxies , we characterize the behavior of the escape fraction in galaxies between 3 \times 10 ^ { 6 } and 3 \times 10 ^ { 9 } solar masses with different spin parameters , amounts of turbulence , and baryon mass fractions . For a given halo mass , escape fractions can vary up to a factor of two , depending on the initial setup of the idealized halo . In a cosmological setting , we find that the time-averaged photon escape fraction always exceeds 25 % and reaches up to 80 % in halos with masses above 10 ^ { 8 } solar masses with a top-heavy IMF . The instantaneous escape fraction can vary up to an order of magnitude in a few million years and tend to be positively correlated with star formation rate . We find that the mean of the star formation efficiency times ionizing photon escape fraction , averaged over all atomic cooling ( T _ { vir } \geq 8000 ~ { } K ) galaxies , ranges from 0.02 for a normal IMF to 0.03 for a top-heavy IMF , whereas smaller , molecular cooling galaxies in minihalos do not make a significant contribution to reionizing the universe due to a much lower star formation efficiency . These results provide the physical basis for cosmological reionization by stellar sources , predominately atomic cooling dwarf galaxies .