We use deep Hubble Space Telescope imaging of the Frontier Fields to accurately measure the galaxy rest-frame ultraviolet luminosity function (UV LF) in the redshift range z similar to 6-8. We combine observations in three lensing clusters, A2744, MACS 0416, and MACS 0717, and their associated parallel. fields to select high-redshift dropout candidates. We use the latest lensing models to estimate the flux magnification and the effective survey volume in combination with completeness simulations performed in the source plane. We report the detection of 227 galaxy candidates at z = 6-7 and 25 candidates at z similar to 8. While the total survey area is about 4 arcmin(2) in each parallel field, it drops to about 0.6-1 arcmin(2) in the cluster core fields because of the strong lensing. We compute the UV LF at z similar to 7 using the combined galaxy sample and perform Monte Carlo simulations to determine the best-fit Schechter parameters. We are able to reliably constrain the LF down to an absolute magnitude of M-UV = -15.25, which corresponds to 0.005 L-star. More importantly, we find that the faint-end slope remains steep down to this magnitude limit with alpha = -2.04(-0.17)(+0.13). We find a characteristic magnitude of M-star = -20.89(-0.72)(+0.60) and log (phi(star)) = -3.54(-0.45)(+0.48). Our results confirm the most recent results in deep blank fields but extend the LF measurements more than two magnitudes deeper. The UV LF at z similar to 8 is not very well constrained below M-UV = -18 owing to the small number statistics and incompleteness uncertainties. To assess the contribution of galaxies to cosmic reionization, we derive the UV luminosity density at z similar to 7 by integrating the UV LF down to an observational limit of M-UV = -15. We show that our determination of log(rho(UV)) = 26.2 +/- 0.13 (erg s(-1) Hz(-1) Mpc(-3)) can be sufficient to maintain reionization with an escape fraction of ionizing radiation of f(esc) = 10%-15%. Future Hubble Frontier Fields observations will certainly improve the constraints on the UV LF at the epoch of reionization, paving the way to more ambitious programs using cosmic telescopes with the next generation of large aperture telescopes such as the James Webb Space Telescope and the European Extremely Large Telescope.