The phase stability and superconductivity of lutetium polyhydrides under pressure were systematically explored via particle swarm optimization. Several lutetium hydrides, such as LuH, LuH3, LuH4, LuH6, LuH8, and LuH12, were found to be dynamically and thermodynamically stable. Combined with the electronic properties, there are a large number of H-s states and low density of Lu-f states at the Fermi level, leading to superconductivity. The phonon spectrum and electron-phonon coupling interaction are considered to calculate the superconducting critical temperature (Tc) of stable lutetium hydrides at high pressure. The new predicted cubic LuH12 has the highest Tc value of 187.2 K at 400 GPa in all the stable LuHn compounds, which was estimated by directly solving the Eliashberg equation. The calculated results provide insights into the design of new superconducting hydrides under pressure.