Recent developments in random geometry

We discuss the models of random geometry that are derived
from scaling limits of large graphs embedded in the sphere and
chosen uniformly at random in a suitable class. The case of
quadrangulations with a boundary leads to the so-called
Brownian disk, which has been studied in a number of recent works.
We present a new construction of the Brownian
disk from excursion theory for Brownian motion indexed
by the Brownian tree. We also explain how the structure
of connected components of the Brownian disk above a
given height gives rise to a remarkable connection with
growth-fragmentation processes.