The remarkable Hubble Space Telescope (HST) data sets from the CANDELS, HUDF09, HUDF12, ERS, and BoRG/HIPPIES programs have allowed us to map the evolution of the rest-frame UV luminosity function (LF) from z~10 to z~4. We develop new color criteria that more optimally utilize the full wavelength coverage from the optical, near-IR, and mid-IR observations over our search fields, while simultaneously minimizing the incompleteness and eliminating redshift gaps. We have identified 5859, 3001, 857, 481, 217, and 6 galaxy candidates at z~4, z~5, z~6, z~7, z~8, and z~10, respectively, from the ~1000arcmin^2^ area covered by these data sets. This sample of >10000 galaxy candidates at z>=4 is by far the largest assembled to date with HST. The selection of z~4-8 candidates over the five CANDELS fields allows us to assess the cosmic variance; the largest variations are at z>=7. Our new LF determinations at z~4 and z~5 span a 6 mag baseline and reach to -16 AB mag. These determinations agree well with previous estimates, but the larger samples and volumes probed here result in a more reliable sampling of >L_{star}_ galaxies and allow us to reassess the form of the UV LFs. Our new LF results strengthen our earlier findings to 3.4{sigma} significance for a steeper faint-end slope of the UV LF at z>4, with {alpha} evolving from {alpha}=-1.64+/-0.04 at z~4 to {alpha}=-2.06+/-0.13 at z~7 (and {alpha}=-2.02+/-0.23 at z~8), consistent with that expected from the evolution of the halo mass function. We find less evolution in the characteristic magnitude M^*^ from z~7 to z~4 the observed evolution in the LF is now largely represented by changes in {phi}_{star}_. No evidence for a non-Schechter-like form to the z~4-8 LFs is found. A simple conditional LF model based on halo growth and evolution in the M/L ratio ({alpha}(1+z}^-1.5^) of halos provides a good representation of the observed evolution.