The advent of nanoscale multilayer (ML) technology has led to great breakthroughs in many scientific and technological fields such as nano-manufacturing, bio-imaging, atto-physics, matter physics and solar physics. ML nanostructures are an enabling technology for the development of mirrors and reflective gratings having high efficiency at normal incidence in the extreme ultraviolet (EUV) range, a spectral region where conventional coatings show a negligible reflectance. In solar physics, ML mirrors have proved to be key elements for both imaging and spectroscopy space instruments, as they allow to make observations of EUV solar plasma emissions with spatial and spectral resolutions never reached before. ML-based instruments have been used in many of the major solar satellites and have flown in numerous sounding rocket experiments; moreover, in the last two decades many studies were performed in order to develop ML structures with increasingly better performance for future solar missions. In this paper, a review of the most promising ML nanostructures developed so far and applied to the observation of solar plasma emission lines is presented. After a brief recall of ML theory, a detailed discussion of the most promising material pairs and layer stack structures proposed and applied to past and current space missions will be presented; in particular, the review will focus on the ML structures having high efficiency in the 6 nm-35 nm wavelength range. Finally, the ML stability to low energy ion bombardment will be discussed.