A turbocharger operable in either single-stage compression mode or two-stage serial compression mode, comprising: a turbine comprising a turbine wheel and a turbine housing, the turbine wheel being affixed to one end of a rotatable shaft, the turbine housing defining an axial bore therethrough in which the turbine wheel is disposed, an exhaust gas inlet, a generally annular turbine volute surrounding the turbine wheel, and a nozzle leading from the turbine volute into the turbine wheel; a compressor comprising a compressor wheel and a compressor housing, the compressor wheel being affixed to an opposite end of the shaft and being disposed in the compressor housing, the compressor wheel comprising a first impeller and a second impeller, the compressor housing defining a first air inlet for leading air into the first impeller, a second air inlet for leading air into the second impeller, a first compressor volute surrounding the first impeller and configured for receiving air pressurized by the first impeller, a crossover duct configured to lead pressurized air from the first compressor volute into the second air inlet, and a second compressor volute surrounding the second impeller and configured for receiving air pressurized by the second impeller, and an air discharge duct configured to lead air from the second compressor volute and out of the compressor housing for delivery to an intake manifold of an internal combustion engine; the compressor housing further defining a bypass duct extending from one of the first compressor volute and the crossover duct, to the air discharge duct; and a bypass valve disposed in the bypass duct, the bypass valve being movable between a closed position and an open position, the closed position preventing pressurized air in the first compressor volute from proceeding along the bypass duct so as to bypass the second impeller, the open position allowing pressurized air in the first compressor volute to proceed along the bypass duct so as to bypass the second impeller and a small amount of air to pass through the crossover duct into the second air inlet and then through the second impeller into the air discharge duct. a turbine comprising a turbine wheel and a turbine housing, the turbine wheel being affixed to one end of a rotatable shaft, the turbine housing defining an axial bore therethrough in which the turbine wheel is disposed, an exhaust gas inlet, a generally annular turbine volute surrounding the turbine wheel, and a nozzle leading from the turbine volute into the turbine wheel; a compressor comprising a compressor wheel and a compressor housing, the compressor wheel being affixed to an opposite end of the shaft and being disposed in the compressor housing, the compressor wheel comprising a first impeller and a second impeller, the compressor housing defining a first air inlet for leading air into the first impeller, a second air inlet for leading air into the second impeller, a first compressor volute surrounding the first impeller and configured for receiving air pressurized by the first impeller, a crossover duct configured to lead pressurized air from the first compressor volute into the second air inlet, and a second compressor volute surrounding the second impeller and configured for receiving air pressurized by the second impeller, and an air discharge duct configured to lead air from the second compressor volute and out of the compressor housing for delivery to an intake manifold of an internal combustion engine; the compressor housing further defining a bypass duct extending from one of the first compressor volute and the crossover duct, to the air discharge duct; and a bypass valve disposed in the bypass duct, the bypass valve being movable between a closed position and an open position, the closed position preventing pressurized air in the first compressor volute from proceeding along the bypass duct so as to bypass the second impeller, the open position allowing pressurized air in the first compressor volute to proceed along the bypass duct so as to bypass the second impeller and a small amount of air to pass through the crossover duct into the second air inlet and then through the second impeller into the air discharge duct.