An illumination system for EUV projection lithography has a beam shaping optical unit for generating an EUV collective output beam from an EUV raw beam of a synchrotron-radiation-based light source. An output coupling optical unit serves for generating a plurality of EUV individual output beams from the EUV collective output beam. In each case a beam guiding optical unit serves for guiding the respective EUV individual output beam toward an object field in which a lithography mask is arrangable. The result is an illumination system with which EUV light of a synchrotron-radiation-based light source is guided to the greatest possible extent without losses and at the same time flexibly.