An optical rectenna--a device that directly converts free-propagating electromagnetic waves at optical frequencies to direct current--was first proposed over 40 years ago, yet this concept has not been demonstrated experimentally due to fabrication challenges at the nanoscale. Realizing an optical rectenna requires that an antenna be coupled to a diode that operates on the order of 1 PHz (switching speed on the order of 1 fs). Diodes operating at these frequencies are feasible if their capacitance is on the order of a few attofarads, but they remain extremely difficult to fabricate and to reliably couple to a nanoscale antenna. Here we demonstrate an optical rectenna by engineering metal-insulator-metal tunnel diodes, with a junction capacitance of ∼2 aF, at the tip of vertically aligned multiwalled carbon nanotubes (∼10 nm in diameter), which act as the antenna. Upon irradiation with visible and infrared light, we measure a d.c. open-circuit voltage and a short-circuit current that appear to be due to a rectification process (we account for a very small but quantifiable contribution from thermal effects). In contrast to recent reports of photodetection based on hot electron decay in a plasmonic nanoscale antenna, a coherent optical antenna field appears to be rectified directly in our devices, consistent with rectenna theory. Finally, power rectification is observed under simulated solar illumination, and there is no detectable change in diode performance after numerous current-voltage scans between 5 and 77 °C, indicating a potential for robust operation.