We present a robust measurement and analysis of the rest-frame ultraviolet (UV) luminosity function at z=4-8. We use deep Hubble Space Telescope imaging over the CANDELS/GOODS fields, the Hubble Ultra Deep Field and the Year 1 Hubble Frontier Field deep parallel observations. These surveys provides an effective volume of 0.6-1.2 x 10^6 Mpc^3 over this epoch, allowing us to perform a robust search for faint (M_UV=-18) and bright (M_UV < -21) galaxies. We select candidate galaxies using a well-tested photometric redshift technique with careful screening of contaminants, finding a sample of 7446 galaxies at 3.5<z<8.5, with >1000 galaxies at z~6-8. We measure the luminosity function using a Markov Chain Monte Carlo analysis to measure robust uncertainties. At the faint end our results agree with previous studies, yet we find a higher abundance of UV-bright galaxies at z>6, with M* ~ -21 at z>5, different than that inferred based on previous trends at lower redshift. At z=8, a single power-law provides an equally good fit to the UV luminosity function, while at z=6 and 7, an exponential cutoff at the bright-end is moderately preferred. We compare to semi-analytical models, and find that the lack of evolution in M* is consistent with models where the impact of dust attenuation on the bright-end of the luminosity function decreases at higher redshift. We measure the evolution of the cosmic star-formation rate density, correcting for dust attenuation, and find that it declines as (1+z)^(-4.3 +/- 0.5) at z>4, consistent with observations at z>9. Our observations are consistent with a reionization history that starts at z>10, completes at z>6, and reaches a midpoint (x_HII = 0.5) at 6.7<z<9.4. Finally, our observations predict that the abundance of bright z=9 galaxies is likely higher than previous constraints, though consistent with recent estimates of bright z~10 galaxies. [abridged]
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