In 1934, Wilhelm Blaschke’s attention focused on a recent construction in metric geometry proposed by Dan Barbilian as a generalization of various models of hyperbolic geometry. It was the year when S.-S. Chern started his doctoral program under Blaschke’s supervision in Hamburg and when in several academic centers in Europe scholars were interested in generalizations of Riemannian geometry. Introduced originally in 1934, Barbilian’s metrization procedure induces a distance on a planar domain through a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian generalized this process to more general domains. In our discussion we plan to show that these spaces are naturally related to Gromov hyperbolic spaces. In several works written with W.G. Boskoff, we explore this connection. We conclude our talk by stating several open problems related to this content.