The NATO Advanced Study Institute "Cosmological Aspects of X-Ray Clus ters of Galaxies" took place in Vel en, Westphalia, Germany, from June 6 to June 18, 1993. It addressed the fruitful union of two topics, cosmology and X-ray clus ters, both of which carry substantial scientific weight at the beginning of the last decenium of the last century in the second millenium of our era. The so far largest X-ray "All-Sky Survey," observed by the ROSAT X-ray satel lite, and ROSAT's deep pointed observations, have considerably enlarged the base of X-ray astronomy, particularly concerning extragalactic sources. Cosmology has gained significant impetus from the large optical direct and spectroscopic surveys, based on high quality 2-dimensional receivers at large telescopes and powerful scan ning devices, harvesting the full information 1 content from the older technique of employing photographic plates. Radioastronomy and IR-astronomy with IRAS, as well as r-astronomy with GRO, continue and strengthen the role of extragalactic research. The rapidly growing computer power in data reduction and data storage facilities support the evolution towards large-number statistics. A most significant push was given to early cosmology by the needs of physics in trying to unravel the nature of forces which govern our material world. The topic of the ASI was chosen because it opens new vistas on this for ever new problem: the universe. Clusters of galaxies probe large-scale matter distributions and the structure of space-time."