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We use the ACS BViz data from the HUDF and all other deep HST ACS fields(including the GOODS fields) to find large samples of star-forming galaxies atz~4 and z~5 and to extend our previous z~6 sample. These samples contain 4671,1416, and 627 B, V, and i dropouts, respectively, and reach to extremely lowluminosities (0.01-0.04 L* or M(UV)~-16 to -17), allowing us to determine therest-frame UV luminosity function (LF) and faint-end slope alpha at z~4-6 tohigh accuracy. We find faint-end slopes alpha of -1.73+/-0.05 at z~4,-1.66+/-0.09 at z~5, and -1.74+/-0.16 at z~6 -- suggesting that the faint-endslope is very steep and shows little evolution with cosmic time. We find thatM*(UV) brightens considerably in the 0.7 Gyr from z~6 to z~4 (by ~0.7 mag fromM*=-20.24+/-0.19 to M*=-20.98+/-0.10). The observed increase in thecharacteristic luminosity over this range is almost identical to that expectedfor the halo mass function -- suggesting that the observed evolution is likelydue to the hierarchical coalescence and merging of galaxies. The evolution inphi* is not significant. The UV luminosity density at z~6 is modestly lower(0.45+/-0.09 times) than that at z~4 (integrated to -17.5 AB mag) though alarger change is seen in the dust-corrected star-formation rate density. Wethoroughly examine published LF results and assess the reasons for their widedispersion. We argue that the results reported here are the most robustavailable. The extremely steep faint-end slopes alpha found here suggest thatlower luminosity galaxies play a significant role in reionizing the universe.Finally, we consider recent search results for galaxies at z~7-8 and use themto extend our estimates of the evolution in M* from z~7-8 to z~4.

UV Luminosity Functions at z~4, 5, and 6 from the HUDF and other Deep HST ACS Fields: Evolution and Star Formation History