Pressurized air systems for aircraft and related methods are described herein. An example pressurized air system includes a compressor having a compressor inlet and a compressor outlet. The compressor inlet receives air from a first air source and the compressor outlet supplies pressurized air to an environmental control system (ECS). The pressurized air system includes a turbine having a turbine inlet to receive air from a second air source, a first overrunning clutch operatively coupled between an output shaft of an accessory gearbox and the compressor, the accessory gearbox operatively coupled to a drive shaft extending from an engine of the aircraft, and a second overrunning clutch operatively coupled between the compressor and the turbine. The first and second overrunning clutches enable the accessory gearbox to drive the compressor during a first mode of operation and enable the turbine to drive the compressor during a second mode of operation.