A multi-stage compressor, namely a multi-stage radial compressor or a multi-stage diagonal compressor, having a compressor rotor, the compressor rotor has a rotor shaft and multiple impellers fastened to the rotor shaft which are subjected to inflow in the axial direction and impellers which are subjected to outflow in the radial direction or diagonal direction. Each impeller has a curved inner shroud, a curved outer shroud and multiple curved impeller blades arranged between the inner shroud and the outer shroud. The inner shroud, the outer shroud, and the impeller blades each consists of a fibre composite material. The rotor shaft extends through a recess in the inner shroud of the respective impeller.