Rotary fluid handling apparatus employing a wheel to provide multi-stage compression or expansion. A wheel having a first set of vanes includes a shroud about those vanes with a second set of vanes outwardly of the shroud. One set of vanes provides for low specific speed flow while the other set of vanes provides for high specific speed flow. A transfer passage interconnects the outlet of the first stage with the inlet of the second stage. The difference in temperature between the inlet flow to the system and the outlet flow from the system may be exchanged to increase efficiency. The multi-stage wheel and associated passages may be configured for either compression or turboexpansion.