Abstract: The Gromov–Hausdorff distance is a fundamental tool in Riemanian geometry, due the topology it generates, and also in applied geometry and topological data analysis, as a metric for expressing the stability of the persistent homology of geometric data (e.g. via the Vietoris-Rips filtration). Whereas it is often easy to estimate the value of the distance between two given metric spaces, its precise value is rarely easy to determine. Some of the best estimates follow from considerations actually related to both the stability of persistent homology and to Gromov‘s filling radius. However, these turn out to be non-sharp. In this talk I will describe these estimates and also results which permit calculating the precise value of the Gromov–Hausdorff between certain pairs of spheres (endowed with their geodesic distance). These results involve lower bounds, which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and from matching upper bounds which are induced from specialized constructions of “correspondences” between spheres.