Koebe's conjecture on conformal uniformization by circle domains

A domain in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is a connected open set. A circle domain is a domain whose complementary components are points or closed disks. Two domains are conformally equivalent if there is a conformal map from one onto the other. Koebe's conjecture. Every domain in the Riemann sphere is conformally equivalent to a circle domain. This is a central problem in conformal uniformization and has remained open for almost 120 years.