We report the first quantum degenerate dipolar Fermi gas, the realization of which opens a new frontier for exploring strongly correlated physics and, in particular, quantum liquid crystalline phases. A quantum degenerate Fermi gas of the most magnetic atom 161Dy is produced by laser cooling to 10 μK before sympathetically cooling with ultracold, bosonic 162Dy. The temperature of the spin-polarized 161Dy is a factor T/T(F)=0.2 below the Fermi temperature T(F)=300 nK. The cotrapped 162Dy concomitantly cools to approximately T(c) for Bose-Einstein condensation, thus realizing a novel, nearly quantum degenerate dipolar Bose-Fermi gas mixture. Additionally, we achieve the forced evaporative cooling of spin-polarized 161Dy without 162Dy to T/T(F)=0.7. That such a low temperature ratio is achieved may be a first signature of universal dipolar scattering.