An illumination system is used to illuminate a specified illumination field of an object surface with EUV radiation. The illumination system has an EUV source and a collector to concentrate the EUV radiation in the direction of an optical axis. A first optical element is provided to generate secondary light sources, and a second optical element is provided at the location of these secondary light sources, the second optical element being part of an optical device which includes further optical elements, and which images the first optical element into an image plane into the illumination field. Between the collector and the illumination field, a maximum of five reflecting optical elements are arranged. These optical elements reflect the main beam either gratingly or steeply. The optical axis, projected onto an illumination main plane, is deflected by more than 30° between a source axis portion and a field axis portion. In a first variant of the illumination system, at least an axis portion between at least two of the reflecting optical elements is inclined relative to the illumination main plane. In a second variant of the illumination system, the optical device, in addition to the second optical element includes precisely three further optical elements, i.e. a third optical element, a fourth optical element and a fifth optical element. In this second variant, the optical axis meets the third, fourth and fifth optical elements at an angle of incidence which is greater than 70°. This construction variants make possible either an increase of the EUV throughput of the illumination system for a given size, or a reduction of the size of the illumination system and thus of the associated projection exposure system for a given EUV throughput.