Chandra observations of rich, relaxed galaxy clusters allow the properties of the X–ray gas and the total gravitating mass to be determined precisely. Here, we present results for a sample of the most X–ray luminous, dynamically relaxed clusters known. We show that the Chandra data and independent gravitational lensing studies provide consistent answers on the mass distributions in the clusters. The mass profiles exhibit a form in good agreement with the predictions from numerical simulations. Combining Chandra results on the X–ray gas mass fractions in the clusters with independent measurements of the Hubble constant and the mean baryonic matter density in the Universe, we obtain a tight constraint on the mean total matter density of the Universe, Ωm, and an interesting constraint on the cosmological constant, ΩΛ. We also describe the ‘virial relations’ linking the masses, X–ray temperatures and luminosities of galaxy clusters. These relations provide a key step in linking the observed number density and spatial distribution of clusters to the predictions from cosmological models. The Chandra data confirm the presence of a systematic offset of ca. 40% between the normalization of the observed mass–temperature relation and the predictions from standard simulations. This finding leads to a significant revision of the best–fit value of σ8 inferred from the observed temperature and luminosity functions of clusters.