A cooling system is provided for cooling a turbine of a gas turbine engine. The system has first and second flow paths for guiding cooling air received from the compressor of the engine. The routes of both flow paths bypass the combustor of the engine. The system also has a preswirler for receiving the cooling air at the ends of the two flow paths, swirling the cooling air tangentially to the engine axis, and delivering the swirled cooling air to a rotor of the turbine. The first flow path is routed through a heat exchanger which cools the cooling air guided by the first flow path relative to the cooling air guided by the second flow path.