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 ∼1000 arcmin2 area covered by these data sets. This sample of >10,000 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∗ galaxies and allow us to reassess the form of the UV LFs. Our new LF results strengthen our earlier findings to 3.4σ significance for a steeper faint-end slope of the UV LF at z ∼ 4, with α evolving from α = -2.06 ∓ 0.13 at to z ∼ 7 (and α = -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 . 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(∝(1+ ∼)-1.5 of halos provides a good representation of the observed evolution.