This thesis describes the development and characterization of high thermal and radiation stable Si-based multilayer mirrors for their application in extreme ultraviolet lithographic steppers. Although EUV Lithography (EUVL) is an optical lithography, the reduced wavelength introduces new challenges because all materials are absorbing at 13.5 nm: The application of multilayer mirrors in EUVL requires not only the highest possible normal-incidence reflectivity but also a long-term thermal and radiation stability at operating temperatures. This requirement is most important in the case of the collector mirror of the illumination system close to the EUV source where a short-time decrease in reflectivity is most likely.