The presentations of this NATO Advanced Study Institute centre around X-ray clusters of galaxies and their role in understanding the structure and evolution of the universe. Reminiscences of the beginning of the extra-solar X-ray astronomy some 30 years ago, of subsequent X-ray satellites which, among other discoveries, found galaxy clusters to be an important new class of intrinsically bright X-ray sources, and highlights from the most recent All Sky Survey by ROSAT serve as an introduction. The topics range from X-ray clusters and cluster environments to cluster masses and dynamics, cooling flows and chemical abundances, cluster lensing and the Sunyaev-Sel'dovich effect, cluster formation and evolution to large-scale structure in the universe and cosmological parameters. The reviews, introducing the various topics, generally conclude with presentations of current and future work and are frequently followed by shorter research notes pertaining to ongoing projects. More than 200 figures illustrate the discussions, more than 20 tables and many quotations in the text provide numerical data for almost 100 clusters of galaxies and give new estimates for the cosmological parameters. Together with a sample of equations, this text should constitute a useful collection of empirical and theoretical quantities and relations in extragalactic X-ray astronomy and cosmology.