X-ray Cluster Cosmology from Einstein to eROSITA

Contributions from X-ray observations of clusters of galaxies to cosmological studies include 1) constraints on cosmological parameters derived from comparing the mass functions of clusters or baryon fractions as a function of redshift, which constrain Omega_matter, sigma_8, and the Dark Energy equation of state, 2) constraints on the mass of light neutrinos and the self-interaction cross-section of dark matter, 3) tests of modified gravity models, and 4) improved understanding of the formation and growth of clusters and supermassive black holes over cosmological time, including the formation of cooling cores, AGN feedback and heavy element enrichment of the intracluster medium. This talk will focus on the results of cluster surveys from the current X-ray missions Chandra and XMM-Newton, past missions Einstein and ROSAT, and those expected from the future eROSITA all-sky survey.