Analogous to radio- and microwave antennas, optical nanoantennas are devices that receive and emit radiation at optical frequencies. Until recently, the realization of electrically driven optical antennas was an outstanding challenge in nanophotonics. In this review we discuss and analyze recent reports in which quantum tunneling-specifically inelastic electron tunneling-is harnessed as a means to convert electrical energy into photons, mediated by optical antennas. To aid this analysis we introduce the fundamentals of optical antennas and inelastic electron tunneling. Our discussion is focused on recent progress in the field and on future directions and opportunities.