In 1908, Koebe conjectured that every domain in the two-dimensional sphere is conformally equivalent to a circle domain, i.e., a domain whose complementary components are points or round disks. Koebe himself confirmed this conjecture for finitely connected domains, but it was not until 1993 that He and Schramm confirmed the conjecture in the countably connected case. The full conjecture remains open. We will discuss a version of this problem for metric spaces that are homeomorphic to a domain in the plane. In this setting, which is inspired by geometric group theory and rigidity theory, there is no a priori smooth structure with which to define the notion of conformality. Instead, we employ a similar condition, called quasisymmetry, which is assumed to hold at all scales and has a rich family of intuitive geometric invariants.