**Abstract:** | Models of two-dimensional random geometry are obtained as universal scaling limits in the Gromov-Hausdorff sense of large graphs embedded in the sphere. These models, which include the Brownian sphere, the Brownian disk and the Brownian plane, are also closely related to the quantum surfaces studied by Miller and Sheffield. We will present recent progress in the study of these random metric spaces. In particular we will discuss some remarkable properties of geodesics and mention some open problems. |