Exceptionally deep observations of the distant universe with the Hubble Space Telescope have consistently pushed the frontiers of human knowledge. How deep can we go? What are the faintest and most distant galaxies we can see with the Hubble Space Telescope now, before the launch of the James Webb Space Telescope? This is the challenge taken up by the Frontier Fields, a director's discretionary time campaign with HST and the Spitzer Space Telescope to see deeper into the universe than ever before. The Frontier Fields combines the power of HST with the natural gravitational telescopes of high-magnification clusters of galaxies to produce the deepest observations of clusters and their lensed galaxies ever obtained. I will review the original goals of the Frontier Fields program and its progress over the last several years. In addition to pushing forward the study of the most distant galaxies, the Frontier Fields have been transformative in the study of galaxy clusters and their lensing properties. Finally, I will discuss the prospects for studying galaxies at cosmic dawn with JWST, extremely large ground-based telescopes, and future space missions over the next decade and beyond.