The binary polyhydrides of heavy rare earth lutetium that shares a similar valence electron configuration to lanthanum have been experimentally discovered to be superconductive. The lutetium polyhydrides were successfully synthesized at high pressure and high temperature conditions using a diamond anvil cell in combinations with the in-situ high pressure laser heating technique. The resistance measurements as a function of temperature were performed at the same pressure of synthesis in order to study the transitions of superconductivity (SC). The superconducting transition with a maximum onset temperature (Tc) 71 K was observed at pressure of 218 GPa in the experiments. The Tc decreased to 65 K when pressure was at 181 GPa. From the evolution of SC at applied magnetic fields, the upper critical field at zero temperature {\mu}0Hc2(0) was obtained to be ~36 Tesla. The in-situ high pressure X-ray diffraction experiments imply that the high Tc SC should arise from the Lu4H23 phase with Pm-3n symmetry that forms a new type of hydrogen cage framework different from those reported for previous light rare earth polyhydride superconductors.